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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I )RD UNIVERSITY LIBKAKII S STANFORD UNIVLlJSirr LIUKAUHS STA SITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES ■ STANFORD (JNI lES STANFORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LI Bl STANFORD university libraries STANFORD UNIVERSR STANFORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LIBRARIE LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFOf )RD university LIBRARIES STANFORD UNIVERSITY LIBRARIES -STA \SITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES . 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STANFORD (JN -IBRARIES • STANFORD UNIVERSITY LIBRARIES STANF LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFORD STANFORD UNIVERSITY LIBRARIES . STANFORD UNIVERSITY LIBRO -'""o University LIBRARIES Stanford university LIBP""'ES MATHEMATICAL PAPEKS. lonlion: C. J. CLAY & SONS, CAMBBIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. Camfmlige: DEIGHTON, BELL AND CO. !Lrqj}ts : F. A. BROCEHAUS. THE COLLECTED MATHEMATICAL PAPERS OF AETHUE CAYLEY, Sc.D., F.E.S., 8ADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE. VOL. II. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1889 [All Rights reserved.^ 9y^i_\o^^ CAMBRIDGE : PRINTED BY C. J. CLAT, ILA. AMD SONS, AT THE UNIVEBSITY PRESS. ADVEETISEMENT. THE present volume contains fifty-eight papers (numbered 101, 102,..., 158) originally published, all but two of them, in the years 1851 to 1860: they are here reproduced nearly but not exactly in chronological order. The two excepted papers are 142, Numerical Tables Supplementary to Second Memoir on Quantics, now first published (1889); and, 143, Tables of the Covariants M to W of the Binary Quintic : from the Second, Third, Fourth, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics (arranged in the present form, 1889) : the determination of the finite number, 23, of the covariants of the quintic was made by Gordan in the year 1869, and the calculation of them having been completed in my Ninth and Tenth Memoirs, it appeared to me convenient in the present republication to unite together the values of all the covariants : viz. those of A to L are given in the Second Memoir 141, and the remainder M to W in the paper 143. I have added to the Third Memoir 144, in the notation thereof, some formulae which on account of a difference of notation were omitted fi:om a former paper, 35. I remark that the present volume comprises the first six of the ten Memoirs on Quantics, viz. these are 139, 141, 144, 155, 156 and 158. I have, in the Notes and References, inserted a discussion of some length in reference to the paper 121, Note on a Question in the Theory of Probabilities: and also some remarks in reference to the theory of Dis- tance developed in the Sixth Memoir on Quantics, 158. C. II. CONTENTS. PAOB 101. Notes on Lagrange's Theorem ....... 1 Gamb. and DubL Math. Jour. t. vi. (1851), pp. 37 — 45 102. On a Double Infinite Series ....... 8 Camb. and DubL Math. Jour, t vi. (1851), pp. 45 — 47 103. On Certain Definite Integrals 11 Camb. and DubL Math. Jour, t vl (1851), pp. 136—140 104. On the Theory of Permutants 16 Camb. and DubL Math. Jour, t vii. (1852), pp. 40 — 51 105. Correction to the Postscript to the Pa/per on Permutants 27 Camb. and DubL Math. Jour, t vii. (1852), pp. 97—98 106. On the Singularities of Surfaces 28 Camb. and DubL Math. Jour. t. vii. (1852), pp. 166—171 107. On the Theory of Skew Surfaces 33 Camb. and DubL Math. Jour, t vii. (1852), pp. 171—173 108. On certain Multiple Integrals connected with the Theory of Attractions 35 Camb. and DubL Math. Jour. t. vii. (1852), pp. 174—178 109. On the Rationalisation of certain Algebraical Equ/Uions . 40 Camb. and DubL Math. Jour. t. viii. (1853), pp. 97—101 110. Note on the Transformation of a TrigonomstHcal Expression . 45 Camb. and DubL Math. Jour. t. ix. (1854), pp. 61 — 62 111. On a Theorem of M. Lejeune-Dirichlet' s 47 Camb. and DubL Math. Jour. t. ix. (1854), pp. 163—165 &2 VUl CONTENTS. PAOB 112. Demonstration of a Theorem relating to the Products of Sums of Squares 49 Phil. Mag. t IV. (1852), pp. 516—519 113. On the Geometrical Representation of the Integral Idx-^Jix + a) {x + h) {x + c) 53 Phil. Mag. t. V. (1853), pp. 281—284 114. Analytical Researches connected with Steiner's Extension of MalfattHs Problem 57 Phil. Trans, t cxlii. (for 1852), pp. 253—278 115. Note on the Porism of the In-and-circum^crihed Polygon . 87 Phil. Mag. t. VI. (1853), pp. 99—102 116. Correction of two Theorems rekuing to the In-and-circum^ scribed Polygon ......... 91 Phil. Mag. t VI. (1853), pp. 376—377 117. Note on the Integral \dx'^J{m'-x){x + a){x + h){x + c) . . 93 Phil. Mag. t VL (1853), pp. 103—105 118. On the Harmonic Relation of two Lines or two Points . . 96 Phil. Mag. t. VI. (1853), pp. 105—107 119. On a Theorem for the Development of a Factorial . . . 98 Phil. Mag. t. VI. (1853), pp. 182—185 120. Note on a Generalisation of the Binomial Theorem . . 101 Phil. Mag. t VI. (1853), p. 185 121. Note on a Question in the Theory of Prohahilities . . . IDS Phil. Mag. t VI. (1853), p. 259 122. On the nomographic Transformation of a Surface of tJie Second Order into Itself ....... . . 10 ^^^ Phil. Mag. t'vi. (1853), pp. 326—333 123. On the Geometrical Representation of an Abelian Integral . 11 — PhiL Mag. t vi. (1853), pp. 414—418 124. On a Property of the Caustic by Refraction of the Circle . 1 Phil. Mag. t VL (1853), pp. 427—431 CONTENTS. IX 125. On the Theory of Groups as depending on the Symbolical . Equation ^=1 Phil. Mag. t. VII. (1854), pp. 40—47 126. On the theory of Groups as depending on the Symbolical Equation ^ = 1. Second Part Phil. Mag. t VII. (1854), pp 408—409 127. On the nomographic Transformation of a Surface of the Second Order into itself PhU. Mag. t VII. (1854), pp. 208—212: continuation of 122 128. Developments on the Porism of the In-and-drcumsciibed Polygon PhiL Mag. t. vii. (1854), pp. 339—345 129. On the Porism of the In-and-circuToscribed Triangle^ and on an irrational Transformation of two Ternary Quadratic Forms each into itself. ....... Phil. Mag. t. IX. (1855), pp. 513—517 130. Deuxihne MSmoire sur les Fonctions doublement PSriodiques . Liouville, t. xix. (1854), pp. 193—208 : sequel to 25 131. Nouvelles Recherches sur les Covariants CreUe, t xlvii. (1854), pp. 109—125 132. RSponse d, une Question proposSe par M. Steiner Crelle, t l. (1855), pp. 277—278 133. Sur un Thiorhme de M. Schlafli ...... Crelle, t. l. (1855), pp. 278—282 134. Remarques sur la Notation des Fonctions Algebriques Crelle, t. l. (1855), pp. 282—285 135. Note sur les Covariants d'une Fanction Quadratique^ OubiquCy ou Biquadratique d, deux IndSterminSes .... Crelle, t l. (1855), pp. 285—287 136. Sur la Transformation d'une Fonction Quadratique en elle- mime par des Substitutions Ihieaires ..... Crelle, t l. (1855), pp. 288—289 137. Seeker ches Ult&i^ieures sur les Determinants gauches Crelle, t. l. (1855), pp. 299—313: continuation of 52 and 69. PAOB 123 131 133 138 145 150 164 179 181 185 189 192 202 CONTENTS. PAOB 138. Recherches sur les Matrices dont hs termes sont des fonctions linSaires d^une seule Ind4tei^min4e . . . . . 216 Crelle, t. l. (1855), pp. 313—317 139. An Introductory Memoir on Quantics 221 Phil. Trans, t cxLiv. (for 1864), pp. 244—258 140. Researches on the Partition of Numbers 235 Phil. Trans, t. cxlv. (for 1855), pp. 127—140 141. A Second Memoir on Quantics 250 Phil. Trans, t. cxlvi. (for 1856), pp. 101—126 142. Numerical Tables Supplementary to Second Memoir on Quantics 276 Now first published (1889) 143. Tables of the Covariants M to W of the Binary Quintic : from the Second, Third, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics 282 Arranged in the present form, 1889 144. A Third Memmr on Quantics 310 PhiL Trans, t. cxlvi. (for 1856), pp. 627—647 145. A Memoir on Caustics 336 Phil. Trans, t. cxlvii. (for 1857), pp. 273—312 146. A Memoir on Curves of the Third Order .... 381 Phil. Trans, t cxlvii. (for 1857), pp. 415—446 147. A Memoir on the Symmetric Functions of the Roots of an Equation 417 Phil. Trans, t. cxlvii. (for 1857), pp. 489 — 496 148. A Memxdr on the Resultant of a System of two Equations . 440 Phil. Trans, t. cxlvii. (for 1857), pp. 703—715 149. On the Symmetric Functions of the Roots of certain Systems of two Equations ......... 454 Phil. Trana t. cxlvii. (for 1857), pp. 717—726 150. A Memoir on the Conditions for the Existence of given Systems of Equalities among the Roots of an Equation . . . 465 Phil. Trans, t. cxlvil (for 1857), pp. 727—731 CONTENTS. XI PAOX 51. Tables of the Sturmian Functions for Equations of the Second^ Thirds Fourth^ and Fifth Degrees 471 Phil. Trans, t cxLVii. (for 1857), pp. 733—736 52. A Memoir on the Theory of MatHces 475 Phil. Trans, t. cxLViii. (for 1858), pp. 17—37 )3. A Memoir on the Automorphic Linear Transfoi^mation of a Bipartite Quadric Function 497 Phil. Trans, t. cxlviii. (for 1858), pp. 39—46 )4. Supplementary Researches on tJie Partition of Numbers . . 506 PhiL Trans, t. cxlviii. (for 1858), pp. 47—52 )5. A Fourth Memoir on Qualities 513 Phil. Trans, t. cxlviii. (for 1858), pp. 415—427 )6. A Fifth Memoir on Qiiantics . . . 527 Phil. Trans, t. cxlviil (for 1858), pp. 429—460 )7. On the Tangential of a Cubic 558 PhiL Trans, t. cxlviil (for 1858), pp. 461—463 8. A Sixth Memoir on Quantics 561 Phil. Trans, t cxLix. (for 1859), pp. 61—90 and References .......... 593 CLASSIFICATION. Geometry Theory of Distance, 158 Surfaces, 106, 107 Transformation of Qnadric Surfaces, 122, 127, 129, 136, 153 Steiner's extension of Mal&tti's Problem, 114 In-and-circumscribed triangle and polygon, 115, 116, 128, 129 Harmonic relation of two lines or points, 118 Question proposed by Steiner, 132 Caustics, 124, 145 Cubic Curves, 146, 157 Analysis Skew Determinants, 137 Attractions and Multiple Integrals, 108 Definite Integrals, 103 Elliptic and Abelian Integrals, 110, 113, 117, 123, 130 Covariants, Quantics <fec., 131, 134, 135, 139, 141, 142, 143, 144, 155, 156, 158 Matrices, 138, 152 Partition of Numbers, 140, 154 Symmetric Functions dec., 147, 148, 149, 150 Lagrange's Theorem, 101 Double Infinite Series, 102 Permutants, 104, 105 Rationalisation of Algebraic Expression, 109 Transformation of Trigonometrical Expression, 110 Theorem of Lejeune-Dirichlet's, 111 Products of Sums of Squares, 112 Factorials, 119 Generalisation of Binomial Theorem, 120 Question in Probabilities, 121 Groups, 125, 126 Theorem of Schlafli's, on Elimination, 133 Sturmian Functions, 151 101] 101. NOTES ON LAGRANGE'S THEOREM. [From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 37 — 45.] I. If in the ordinary form of Lagrange's theorem we write {x + a) for x, it becomes X = hf(a + x), F(a + x)=^Fa+jF'afa + &c (1) It follows that the equation F{a + x) = Fa + \j~^^{F'a/a)-i- (2) must reduce itself to an identity when the two sides are expanded in powers of x\ A da (3) or writing for shortness F, f instead of Fa, fa, and S for -r- , we must have (where p extends from to r). Or what comes to the same, [r]--^^°M p[P->?-[r-p]-^[>-ir' ^-^-^-^"-^^} ^*> where s extends from to (^ — p). The terms on the two sides which involve Z^F are immediately seen to be equal ; the coefficients of the remaining terms S'jP on tht^ second side must vanish, or we must have o. II, 1 s.^ 2 NOTES ON Lagrange's theorem. [101 (s being less than r). Or in a somewhat more convenient form, writing p, q and it- for p — 8, r—p and r — «, where 8 is constant and p and q vary subject to p+q = ky k being a given constant diflferent from zero (in the case where A:=0, the series reduces itself to the single term - ). The direct proof of this theorem will be given presently. II. The following symbolical form of Lagrange's theorem was given by me in the Mathematical Journal, vol. ill. [1843], pp. 283—286, [8]. If x^a^-hfx, (7) then Suppose /r = ^ (6 + h^sc\ or a? = a + A^ (6 + ky^x), then Fx= (j-Y'^'^F'a c*«<*+*^». But (In fact the two general terms {<t>{b + kita)}'^ and (^)*^e*^(<^)"', of which the former reduces itself to e db(if)by^, are equal on account of the equiva- lence of the symbols e**"^ and (A)*^e*^). Hence X = a •\- h4> (b •\- kyfrx), (8) ^-=(i)*''"(^)'"^'«^*"'"^ and the coefficient of h^k^ is w r^(^r^'«<^'^)"(s)"('^>"- 101] NOTES ON Lagrange's theorem. 3 A similar formula evidently applies to the case of any finite number of functions 0, y^, &c. : in the case of an infinite number we have or the coefficient of h^kH^ ... is mrmrY ■ :. (r«)" '" ■ {mT <♦')- ■ (s)' <+'>-• ■ • the last of the series m, n, p being always zero; e.g. in the coefiicient of h^k^, account must be had of the factor f-i-j ('^V* or (-^c)**. The above form is readily pn)ved independently by Taylor s theorem, without the assistance of Lagrange's. If in it we write h = k, &c., a = 6 = &c., and ^ = '^ = &c. =/, we have -F (a + hf{a + hf{a +...) = Fx, where ar = a + hfx. Hence, comparing the coefficient of A* with that given by Lagrange's theorem, whtre m + w+&c. = «, and as before Fa, /(/, .- have been replaced by F^ / 8. By comparing the coefficients of h^F, where n ■¥p-\-... =<, the last of the series n, p ,., always vanishing. The formula (10) deduced, as above mentioned, from Taylors theorem, and the subsequent formula (11) with an independent demonstration of it, not I believe materially different from that which will presently be given, are to be found in a memoir by M. Collins (volume ii. (1H38) of the Memoirs of the Academy of St Petersburg), who appears to have made very extensive researches in the theory of developments as connected with the combina- torial analysis. in. To demonstrate the formula (6), consider, in the first place, the expression s^^^,[m''^')m-^-^)\< ii^here j9 + 5 = it. Since 1_ _1 / 1 1_ __\ 1—2 4 NOTES ON Lagrange's theorem. [101 this is immediately transformed into = J 5 ^-j?j^j, {^ (p + 1) (p + » + !)(«>' .y>*'8/)(8«/-'--»-.) iu which last expression p + 9 = (^ — 1 ). Of this, after separating the factor Sf, the general term is [/>?[?-«]*- *p <^ + * + ^ <^-^> (^^z-^^-) equivalent to I f^ «' V- « f^5f {* (p + a + 1) (p + * + a + 1) (SV^+'+-) (S^/-^— -^0 - *p (p + « + ^ (s^y*^*) W'"'''^')]* in which last expression p + g=A;-a — 1. By repeating the reduction j times, the general term becomes 1 1 A;(ifc-a-l)(ifc-a-/8-2) ... [a]*|j8]^... s-+»/.y +'/... x[p + » + ^ + a + /S...+j-ip-^ (SPfP+'-^-^fi") {B^f-P • » i - ^ •)}, where the sums a + /3... contain / terms,/ being less than j or equal to it, and S extends to all combinations of the quantities a, /3... taken / and / together (so that the summation contains 2^ terms). Also p4-3 = A; — a — )8... (J terms) — j, and the products A; (A; - a - 1 ) (Jfc - o - )8 - 2) . . . and [a]* [fif, . . . S*+y . ^ +y. . . contain each of them j terms. Suppose the reduction continued until A: — a — )8 . . . ( j terms) — j = 0, then the only values of p, q are p = 0, ? = ; and the general term of becomes 1 1 8"+»/" ifi+^f t'-i-* kik-a-l){k-a-^-2)... [aj« [/SJ". . . •'•* •'-•^ I] NOTES ON Lagrange's theorem. If ^ =s 0, the general term reduces itself to mee finally, if ^ = — - , the general term of omes 1 it is readily shown that the sum contained in this formula vanishes, which proves equation in question. IV. The demonstration of the equation (11) is much simpler. We have t is, "e n extends from n = 1 to n = ^. Similarly &C. ence, substituting successively, and putting t — n—p — q = r, &c., d the last of these corresponding to a zero value of the last of the quantities . is evidently the required equation (11). V. formula (18) in my paper on Lagrange's theorem (before referred to) is incorrect. at present, after giving the proper form of the formula in question, to the result of the substitution indicated at the conclusion of the paper. It onvenient to call to mind the general theorem, that when any number 6 NOTES ON LAGRANQJi's THEOREM. [lOl of variables a?, y, z .,. are connected with as many other variables t/, r, ti; . . . by the same number of equations (so that the variables of each set may be considered as functions of those of the other set) the quotient of the expressions dxdy ... and dudv ... is equal to the quotient of two determinants formed with the functions which equated to zero express the relations between the two sets of variables ; the former with the differential coefficients of these functions with respect to a, » . . . , the latter with the differential coefficients with respect to x, y Consequently the notation ^-T may be considered as representing the quotient of these determinants. This being premised, if we write X'-u — h0(x, y . . .) = 0, y-V'-k<f>(x, y ...) = 0, then the formula in question is if for shortness the letters 0, 0, . . . , F denote what the corresponding functions become when M, v, ... are substituted for x, y, — Let -r denote the value which , j '" , A dudv... considered as a function of x, y . . . , assumes when these variables are changed into w, V, . . . , we have V = 1-AM, -AS^d... !. — KOu^» 1 — A?Ov0 . . . By changing the function F, we obtain Fix, y...) = S«***S^*^*...«*^**- .FV; where, however, it must be remembered that the A, A:, ... , in so far as they enter into the function V, are not aflfected by the symbols AS^, AS*,... In order that we may consider them to be so affected, it is necessary in the function V to replace A, A:, &c. h k by ^ , ^ , &c. Also, afler this is done, observing that the symbols ASu^, hB„0 ... affect Ou Off a function gW+u-h ... /^ xh^ symbols hBuO, hB„0,.,. may be replaced by S,/, S/, ..., where the is not an index, but an affix denoting that the differentiation is only to be performed with respect to u, t; ... so far as these variables respectively enter into the function 0. Transforming the other lines of the determinant in the same manner, and taking out from Su "K * ••• the factor SuB^ ... in order to multiply this last factor into the determinant, we obtain Fij-, y...) = S„"»- S."'-' . . . ««-*♦• F D ; where n= ««-««*. -««♦.... , 101] NOTES ON Lagrange's theorem. 7 in which expression S„, S„... are to be replaced by The complete expansion is easily arrived at by induction, and the form is somewhat singular. In the case of a single variable u we have □ = Su, in the case of two variables, □ = S„'S„' + S„'S„^ + K^K^- Or writing down only the aflSxes, in the case of a single variable we have F] in the case of two variables FF, Fd, <f)F] and in the case of three variables FFF, <f>FF, x^^y ^X^> ^^^> ^^^. ^^<^» ^^^> ^H* ^X^» 0^0. X-F<^, 4>F0, xxF^ ^X^> X^^'i where it will be observed that 6 never occurs in the first place, nor ^ in the second place, nor d, <f> (in any order) in the first and second places, &c., nor 0, (f>, x 0^ *^y order) in the first, second, and third places. And the same property holds in the general case for each letter and binary, ternary, &c. combination, and for the entire system of letters, and the system of affixes contains every possible combination of letters not excluded by the rule just given. Thus in the case of two letters, forming the system of aflBxes FF, F0, tf>F, 0F, F<\>, 0<f>, <f)0, the last four are excluded, the first three of them by containing in the first place or <f> in the second place, the last by containing <l>, in the first and second places : and there remains only the terms FF, F0, ^F forming the system given above. Substituting the expanded value of □ in the expression for F (a;, y...), the equation may either be permitted to remain in the form which it thus assumes, or we may, in order to obtain the finally reduced form, after expanding the powers of A, A: . . . , connect the symbols S^*, S„*...Su', &c. with the corresponding functions 0, <f).,.F, and then omit the affixes ; thus, in particular, in the case of a single variable the general term of Fx is (the ordinary form of Lagranges theorem). In the case of two letters the general term of F{xy y) is (see the MScaniqvs Celeste, [Ed. 1, 1798] t. i. p. 176). In the case of three variables, the general term is MWW' ^'^'^''"*"''"' {^*V«AS^ +...}. the sixteen terms within the { } being found by comparing the product S„SpS«, with the system FFF, 4>FF^ &c., given above, and then connecting each symbol of diff'eren- tiation with the function corresponding to the aflfix. Thus in the first term the ^> ^vt ^vi ^a^b affect the F, in the second term the h^ affects ^^, and the 8^ and h^ each affect the F, and so on for the remaining terms. The form is of course deducible from Laplace's general theorem, and the actual development of it is given in Laplace's Memoir in the Hist, de VAcad. 1777. I quote from a memoir by Jacobi which I take this opportunity of referring to, "De resolutione equationum per series infinitas," CreUe, t. vi. [1830], pp. 257 — 286, founded on a preceding memoir, "Exercitatio algebraica circa discerptionem singularem fractionum quae plures variabiles involvunt," t. V. [1830], pp. 344—364. Stone Buildings, April 6, 1850. 8 [102 102. ON A DOUBLE INFINITE SERIES. [Fn)m the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 45 — 47.] The following completely paradoxical investigation of the properties of the fianction r (which I have been in possession of for some years) may perhaps be found interesting from its connexion with the theories of expansion and divergent serieis. Let Sr^r denote the sum of the values of ifyr for all integer values of r from — X to X . Then writing w = 2^[n-l]'-a:«-»-^, (1) (where n is any number whatever), we have immediately ^'- = 2r [n - ly-^^ af-*^ ^-^rin- If x^"'-'' = u ; 1 . du ^ that IS, ^ ~ ^» ^' ^ ^ ^»»^» (the constant of integration being of course in general a function of n). Hence (7n^ = 2r[n-l]'*ic~-^-^; (2) or 6* is expanded in general in a doubly infinite necessarily divergent series of /fractional powers of x, (which resolves itself however in the case of n a positive or negative integer, into the ordinary singly infinite series, the value of Cn in this case being immediately seen to be Fn). The equation (2) in its general form is to be considered as a definition of the function (?». We deduce from it Xr [n - 1]'- (flur)'»->-^ = C„e«* , 1^ [n' - ly {aaff-^-^' = Cn'^ ; 102] ON A DOUBLE INFINITE SERIES. 9 and also 2jk [n + n' . . . - 1]* {a (a? + a?' . . . )}«+n'...-i-* = (?«+«•... c«<«+-^ •>. Multiplying the first set of series, and comparing with this last, Cn+n' ...2.. ^ ... [n - l]*- [n' ^ly... a;~-^- a/«-»-^ . . . = CnCn'...[n + n'...-l]*(a; + a;'. ..)•'"'**'•""'"*, (3) (where r, r denote any positive or negative integer numbers satisfying r + r'+...=A:+l— p, p being the number of terms in the series n, n\...). This equation constitutes a multinomial theorem of a class analogous to that of the exponential theorem contained in the equation (2). In particular C^n' ... 2,y ... [n-iy [n' - ly . . . = CnC^. . . . [n + n' . . . - l]*|,»+«'...-i-t, (4) and if p = 2, writing also m, n for n, n', and k—l — r for r*, C„+,2,[m-l]'[«-l]*-'-'=0„C„[m + n-l]*2»+"-'-* (5) or putting k = and dividing, C„(7„-C„+„ = 2„^-2,[m-l]'[n-l]-'-'. (6) Now the series on the second side of this equation is easily seen to be convergent (at least for "positive values of m, n). To determine its value write then (m, n) = I af^^ (1 — w)^^ dx ; J F(m, n)= I a?^»(l~ir)«-*(ir+ f af"-^ {I - xf^^ dx \ J Q J Q and by successive integrations by parts, the first of these integrals is reducible to 1 ^_^-^Y 2r [w — l]*" [^ — l]"^"', ^ extending from — 1 to — x inclusively, and the second to ^^li^^^zi ^r [^ — l]*" [w — l]""*~^i ^* extending from to oo ; hence or C„C„-.C^„ = F(m, n) (7) C. 11. 2 10 ON A DOUBLE INFINITE 8ERI1*>$. [102 which proves the identity of Cn^ with the function T (m), {Substituting in two of the preceding equations, we have TnTn'. . . - T (n + r' . . .) = [„+„'... _l]fcyn-^n..-.-t ^ry... [n - 1]' [n' - 1]' (8) (where, as before, p denotes the number of terms in the series n, n\... and r+r'4-...=ifc+l— /)), the first side of which equation is, it is well known, reducible t-o a multiple definite integral by means of a theorem of M. Dirichlet's. And ''("'• ") = [,« + n-iy 2"^»— * ^' ^"' - ^^' t» - 1]*— ^ (9) , where r extends from — x to 4- « , and k is arbitrary. By giving large negative values to this quantity, very convergent series may be obtained for the calculation of F(m, n)]. 103] 11 103. ON CERTAIN DEFINITE INTEGRALS. [From the Cambridge and Dvblin Mathematical Journal, vol. vi. (1851), pp. 136 — 140.] Suppose that for any positive or negative integral value of r, we have -^(rc + ra) = Ur yp'Xy Ur being in general a function of x, and consider the definite integral J — 00 'a being any other function of x. In case of either of the functions yp'x, 'Vx becoming iTiiinite for any real value a of x, the principal value of the integral is to be taken, that is, 7 is to be considered as the limit of (j + f* '^irx^xdx, (€ = 0), a»n.d similarly, when one of the functions becomes infinite for several of such values We have / r(r+i)a \ / = (... I +...j^a?^«da?; ^^ changing the variables in the different integrals so as to make the limits of each ^» O, we have 1=1 [2'^(ic + ra)^(a; + ra)]da?, Jo "^ Extending to all positive or negative integer values of r, that is, I=ryftx[XUr'ir{x + ra)]dx, (A) Jo 2—2 i 12 ON CERTAIN DEFINITE INTEGRALS. [103 which is true, even when the quantity under the integral sign becomes infinite for particular values of x, provided the integral be replaced by its principal value, that is, provided it be considered as the limit of or where a, or one of the limiting values a, 0, is the value of x, for which the quantity under the integral sign becomes infinite^ and 6 is ultimately evanescent. In particular, taking for simplicity a = tt, suppose -^ (a? 4- tt) = ± -^x, or -^ (a? + nr) = {±y y^x ; then observing the equation Z — - — = cot X, or = cosec x, x + nr according as the upper or under sign is taken, and assuming 'Vx = x~*^, we have finally the former equation corresponding to the case of -^ (j? + tt) = y^x^ the latter to that of '^ (ic 4- tt) = — y^x. Suppose y^^x = yp'gx, g being a positive integer. Then r * yjr^xdx _ ^ f * y^xdx also if >fr (j; + -w) = i^jr, then -^^ (;c + tt) = >^,a: ; but if -^/r (.-c + tt) = — i^J-, then '^^{x-^-ir) = ±ylr^x, the upper or under sign according as g is even or odd. Combining these equations, we have y^(x-{-7r) = yp'X, g even or odd, •^ (a; 4- tt) = — '^a?, g even. ' cot x] At = < j,^ - j' ^x [(J^J" ' cot x] .ir ; /.: fgxdx _(^y-^ f '^ (a? 4- "w) = — -^a?, g odd, g^-'^\x -T- 1 cosec /-.^=S>-V-/>-« |-0-«^^^^^-| ^^(..^'IJ^x [(^"'cosecx] d,. 103] ON CERTAIN DEFINITE INTEGRALS. 13 In particular sinxdx f Sin a J -00 *^ / sin ^07 f -T J cot j: cir = (/**"* / sina: (j-j coseca? ch, g even, I sin^ra: f-j-j coseeo: c2a?= 9^"^] '^^^^ (;/") coseca: cir, jr odd, I sin gxcotxdx^Tr, g even, I sin ^a: cosec iTcia? = TT, g odd, C'tSLUxdx ^ « = 0, i&c., the number of which might be indefinitely extended. The same principle applies to multiple integrals of any order: thus for double integrals, if '^(x + ra, y + rb) = Ur ,i'^ (a:, y), then I i ir{x, y)^(x, y)dxdy^r ( fix, y) 2 £7,,, ^ (a? + ra, y + sb). ... (B) J— 3oJ-ao J J In particular, writing WyV for a, 6, and assuming y^(x + rWf y + sv) = (±y {±y yfr (x, y); also '^(x, y) = (x + iy)~'^, where as usual i=\/ — 1» where -a/ . • N x' (±X(±)'l ^ ^^ (x + ty -{-rw + sm) S extending to all positive or negative integer values of r and s. Employing the notation of a paper in the Cambridge Mathematical Journal, "On the Inverse Elliptic Functions," t. iv. [1845], pp. 257 — 277, [24], we have for the diflferent combinations of the ambiguous sign, ^/ • V iS(x+iy) 1 1. -, -, e(a? + ty)= , 7'\ =j./-r'-\> ^ ^^ y{x + %y) ^(a? + ty) 14 ON CERTAIN DEFINITE INTEGRALS. [103 3. + ft / 1 • \ _ g (a? 4- iy ) _ /(a? - f iy) 4. +, +, e / . • \ 7'(^+*y) where ^, /, ^ are in fact the symbols of the inverse elliptic functions (Abel's notation) corresponding very nearly to sin am, cos am, A am. It is remarkable that the last value of cannot be thus expressed, but only by means of the more complicated transcendant yx, corresponding to the H(x) of M. Jacobi. The four cases correspond obviously to 1 . 1^ (a? + rw, y + w) = (-)''+• yjr (», y), 2. -^(x-^-rw, y + 8v) = (-y ^{x, y), 3. ^{x-^-rw, y + 8v) = (-'y y^(x, y\ 4. y}r(x + rw, y + 8v)= y^(x, y). The above formulae may be all of them modified, as in the case of single integrals, by means of the obvious equation The most important particular case is 00 # 00 ^ 00 # /.J- 00 (* + iy) for in almost all the others, for example in the second integration cannot be effected. Suppose next -^{x, y) is one of the functions 7(a? + iy), g{x + iy), G(x-\'iy\ CS (a? + iy), so that ^|r(x-\'rw, y + 8v) = (±y{±yUrjylr{x, y\ where (see memoir quoted). Then, retaining the same value as before of "9 (x, y), we have still the formula (B), in which ^ ^^ X + ly '\-rw + 8m But this summation has not yet been effected; the difficulty consists in the variable factor €^* ('•«'-«*') in the numerator, nothing being known I believe of the decomposition of functions into series of this form. 103] ON CERTAIN DEFINITE INTEGRALS. 15 On the subject of the preceding" paper may be consulted the following memoirs by Raabe, "Ueber die Summation periodischer Reihen," Crelle, t. xv. [1836], pp. 355 — 364, and •* Ueber die Summation harmonisch periodischer Reihen," t. xxiil. [1842], pp. 105 — 125, and t. xxv. [1843], pp. 160 — 168. The integrals he considers, are taken between the limits 0, oo (instead of — oo , oo ). His results are consequently more general than those given above, but they might be obtained by an analogous method, instead of the much more complicated one adopted by him : thus if <^ (a? -h 2'rr) = <f)x, the integral / 90 if>x — reduces itself to ^•r*'?Ssr-r'^*'[i+^"(iA™-2iy • provided I dx(t>x = 0. The summation in this formula may be effected by means of Jo the function F and its differential coefficient, and we have /: , dx ^ X ^'^) which is in effect Raabe's formula (10), Crelle, t. xxv. p. 166. By dividing the integral on the right-hand side of the equation into two others whose limits are 0, tt, and tt, 27r respectively, and writing in the second of these 27r — a: instead of x, then J ^ 29r j J <l>x i:^-h<^(27r-a:)— ^ 27r>' ^ '•'£) f '>-£)' dx; or reducing by ■- (4) '■'('- s) — TT cot ^X, we have --^^ j ^ — =ij <l>xcoHxdx-^j [4>x -h (f> (in - x)] — JT^» ^"ich corresponds to Raabe's formula (10'). If <^ (- aj) = - <f>x, so that ^ -h <^ (27r — a?) = 0, ^^^ last formula is simplified ; but then the integral on the first side may be replaced k f * dx "Ml <l>x — ,80 that this belongs to the preceding class of formulse. </ — ae X 16 [104 104. ON THE THEORY OF PERMUTANTS. [From the Cavibridge and Dublin Maihematical Journal, vol. vii. (1852), pp. 40 — 5l.] A FORM may by considered as composed of blanks which are to be filled up by inserting in them specializing characters, and a form the blanks of which are so filled up becomes a symbol. We may for brevity speak of the blanks of a symbol in the sense of the blanks of the form from which such symbol is derived. Suppose the characters are 1, 2, 3, 4,..., the symbol may always be represented in the first instance and without reference to the nature of the form, by F1334... And it will be proper to consider the blanks as having an invariable order to which reference will implicitly be made; thus, in speaking of the characters 2, 1, 3, 4,... instead of as before 1, 2, 4,... the symbol will be V^^,., instead of V^^,.. , When the form is given we shall have an equation such as according to the particular nature of the form. Consider now the characters 1, 2, 3, 4,..., and let the primitive arrangement and every arrangement derivable from it by means of an even number of inversions or interchanges of two characters be considered as positive, and the arrangements derived from the primitive arrangement by an odd number of inversions or interchanges of two characters be considered as negative ; a rule which may be termed " the rule of signs." The aggregate of the symbols which correspond to every possible arrangement of the characters, giving to each symbol the sign of the arrangement, may be termed a ** Permutant ; " or, in distinction from the more general functions which will presently be considered, a simple permutant, and may be represented by enclosing the sjTnbol in brackets, thus {V^^,,,). And by using an expression still more elliptical than the blanks of a symbol, we may speak of the blanks of a permutant, or the characters of a permutant. 104] ON THE THEORY OP PERMUTANT8. 17 As an instance of a simple permutant, we may take (r^)^r^+ v^ + r^,^ f«- f«- f„; and if in particular Fia = ai6j(J», then It follows at once that a simple permutant remains unaltered, to the sign prhs according to the rule of signs, by any permutations of the characters entering into the per- mutant For instance, (F-„) = (F„) = (F„) = - ( F«.) = - (F„) = - (F„). Consequently also when two or more of the characters are identical, the permutant vanishes, thus The form of the symbol may be such that the sjrmbol remains unaltered, to the sign pris according to the rule of signs, for any permutations of the characters in certain particular bUmka Such a system of blanks may be termed a quote. Thus, if the first and second blanks are a quote, and consequently (F„) = 2(F„+F„+F„); and if the blanks constitute one single quote, ( ^m . . . ) = iV K ug . . . , where iV=1.2.3...w, n being the number of characters. An important case, which will be noticed in the sequel, is that in which the whole series of blanks divide themselves into quotes, each of them containing the same number of blanks. Thus, if the first and second blanks, and the third and fourth blanks, form quotes respectively, It is easy now to pass to the general definition of a "Permutant." We have only to consider the blanks as forming, not as heretofore a single set, but any number of distinct sets, and to consider the characters in each set of blanks as permutable inter ae and not otherwise, giving to the sjnnbol the sign compounded of the signs corresponding to the arraDgements of the characters in the different sets of blanks. Thus, if the first and second blanks form a set, and the third and fourth blanks form a set, The word 'set' will be used throughout in the above technical sense. The particular mode in which the blanks are divided into sets may be indicated either in words or by some superadded notation. It is clear that the theory of permutants depends ultimately on that of simple permutants; for if in a compound permutant we first write down all the terms which can be obtained, leaving unpermuted the characters c. II. 3 18 ON THE THEORY OP PERMUTANT8. [104 of a particular set, and replace each of the terms so obtained by a simple permutant having for its characters the characters of the previously unpermuted set, the result is obviously the original compound permutant. Thus, in the above-mentioned case, where the first and second blanks form a set and the third and fourth blanks form a set ((f;»o)-=(^i~)-(f.^). in the former of which equations the first and second blanks in each of the permutants on the second side form a set, and in the latter the third and fourth blanks in each of the permutants on the second side form a set, the remaining blanks being simply supernumerary and the characters in them unpermutable. It should be noted that the term quote, as previously defined, is only applicable to a system of blanks belonging to the same set, and it does not appear that anything would be gained by removing this restriction. The following rule for the expansion of a simple permutant (and which may be at once extended to compound permutants) is obvious. Write down all the distinct terms that can be obtained, on the supposition that the blanks group themselves in any manner into quotes, and replace each of the terms so obtained by a compound permutant having for a distinct set the blanks of each assumed quote; the result is the original simple permutant. Thus in the simple permutant (Vmd, supposing for the moment that the first and second blanks form a quote, and that the third and fourth blanks form a quote, this leads to the equation ( F;«) = + (( F,^)) + (( r^)) + (( F,«)) + (( F^)) + (( F„,)) + (( F«„)), where in each of the permutants on the second side the first and second blanks form a set, and also the third and fourth blanks. The blanks of a simple or compound permutant may of couree, without either gain or loss of generality, be considered as having any particular arrangement in qaaoe, for instance, in the form of a rectangle : thus F„ is neither more nor less general than Fis4. The idea of some such arrangement naturally presents itself as affording a means of showing in what manner the blanks are grouped into seta But, considering the blanks as so arranged in a rectangular form, or in lines and columns, suppose in the first instance that this arrangement is independent of the grouping of the blanks into sets, or that the blanks of each set or of any of them are distributed at random in the different lines and columns. Assume that the form is such that a sjnnbol '^«^y ... is a function of symbols Vafiy..., Va^y'...t &c. Or, passing over this general case, and the case (of intermediate generality) of the function being a symmetrical 'function, assume that afi^y,.. 104] ON THE THEORY OF PBRMUTANXa 19 is the product of symbols V^i^y..., V.js'y..., &c Upon this assumption it becomes important to distinguish the different ways in which the blanks of a set are distributed in the different lines and columns. The cases to be considered are : (A). The blanks of a single set or of single sets are situated in more than one column. (E), The blanks of each single set are situated in the same column. (C). The blanks of each single set form a separate column. The case (B) (which includes the case (C)) and the case (C) merit particular consideration. In fact the case (E) is that of the functions which I have, in my memoir on Linear Transformations in the Journal, [13, 14] called hyperdeterminants, and the case ((7) is that of the particular class of hyper- determinants previously treated of by me in the Cambridge PhUoaophical Tranaactiona, [12] and also particularly noticed in the memoir on Linear Transformations. The functions of the case (B) I now propose to call '' Intermutants," and those in the case (C) *' Commutants." Commutants include as a particular case '' Determinants," which term will be used in its ordinary signification. The case (A) I shall not at present discuss in its generality, but only with the further assumption that the blanks form a single set (this, if nothing further were added, would render the arrangement of the blanks into lines and columns valueless), and moreover that the blanks of each line form a quote: the permutants of this class (from their connexion with the researches of Pfaflf on differential equations) I shall term "Pfaffians." And first of commutants, which, as before remarked, include determinants. The general expression of a commutant is (^11 ); or ai ...^ 11... 22 nn 22 nn J and (stating again for this particular case the general rule for the formation of a permutant) if, permuting the characters in the same column in every possible way, considering these permutations as positive or negative according to the rule of signs, one system be represented by '1 " 1 • • • the commutant is the sum of all the different terms The different permutations may be formed as follows: first permute the characters in all the columns except a single column, and in each of the arrangements so obtained permute entire lines of characters. It is obvious that, considering any one of the arrangements obtained by permutations of the characters in all the columns but one, the permutations of entire lines and the addition of the proper sign will only reproduce 3—2 20 ON THE THEORY OP PERMUTANT8. [104 the same 83nnbol — in the case of an even number of columns constantly with the positive sign, but in the case of an odd number of columns with the positive or negative sign, according to the rule of signs. For the inversion or interchange of two entire lines is equivalent to as many inversions or interchanges of two characters as there are characters in a line, that is, as there are columns, and consequently intro- duces a sign compounded of as many negative signs as there are columns. Hence Theoreh a commutant of an even number of columns may be calculated by considering the characters of any one column (no matter which) as supernumerary unpermutable characters, and multiplying the result by the number of permutations of as many things as there are lines in the commutant The mark -f* added to a commutant of an even number of columns will be employed to show that the numerical multiplier is to be omitted. The same mark placed over any one of the columns of the commutant will show that the characters of that particular column are considered as non-permutable. A determinant is consequently represented indiflferently by the notations 11^ t t > + ir 22 • 22 • 22 • • • , KM , • and a commutant of an odd number of cohimns vanishes identically. By considering, however, a commutant of an odd number of columns, having the characters of some one column non-permutable, we obtain what will in the sequel be Hpoken of as commutants of an odd number of columns. This non-permutability will be denoted, as before, by means of the mark f placed over the column in question, and it is to be noticed that it is not, as in the case of a commutant of an even number of columns, indifferert over which of the columns the mark in question is placed; and consequently there would be no meaning in simply adding the mark f to a com- mutant of an odd number of columns. A commutant is said to be symmetrical when the symbols Fo/jy... are such as to remain unaltered by any permutations inter se of the characters a, /8, 7 . . . A com- mutant is said to be skew when each symbol V^py, is such as to be altered in sign only according to the rule of signs for any permutations inter se of the characters a, /9, 7 . . . , this of course implies that the symbol Va^y.,. vanishes when any two of the characters a, 13, 7... are identical. The commutant is said to be demi-skew when Fa,^.y... is altered in sign only, according to the rule of signs for any permutation inter se of non-identical characters a, /8, 7,... An intermutant is represented by a notation similar to that of a commutant. The sets are to be distinguished, whenever it is possible to do so, by placing in contiguity the symbols of the same set, and separating them by a stroke or bar from the symbols 104] ON THE THEORY OP PERMUTANT8. 21 of the adjacent sets. If, however, the symbols of the same set cannot be placed con- tiguously, we may distinguish the symbols of a set by annexing to them some auxiliary character by way of su£Bx or otherwise, these auxiliary symbols being omitted in the final result. Thus ri 1 la) 2 3 4 2 3 3 2b oa 6b would show that 1, 2 of the first column and the 3, 4 of the same column, the 1, 2 and the upper 3 of the second column, and the lower 3 of the same column, the 1, 5 of the third column, and the 2, 6 of the same column, form so many distinct sets, — the intermutant containing therefore (2.2.6.1.2.2 = ) 96 terms. A commutant of an even number of columns may be considered as an intermutant such that the characters of some one (no matter which) of its columns form each of them by itself a distinct set, and in like manner a commutant of an odd number of columns may be considered as an intermutant such that the characters of some one determinate column form each of them by itself a distinct set The distinction of sjnmmetrical, skew and demi-skew applies obviously as well to intermutants as to commutants. The theory of skew intermutants and skew commutants has a connexion with that of Pfa£Bans. Suppose F.^y... = V^+fi+y... (which implies the sjrmmetry of the intermutant or com- mutant) and write for shortness F© = a, Fi = 6, Fj = c, &c. Then \i 1 1 T [j J]=(ac-n &C. The functions on the second side are evidently hyperdeterminants such as are discussed in my memoir on Linear Transformations, and there is no diflSculty in forming directly from the intermutant or commutant on the first side of the equation the symbol of derivation (in the sense of the memoir on Linear Transformations) from which the hyperdeterminant is obtained. Thus ri' is 12 . UU, r]' is liU-oU', 1 1 1 1 is 12 .UU. 12U'U\ 22 ON THE THEORY OF PEBMUTANTS. [104 each pennutable column corresponding to a 12(') and a non-permutable column 1 1 changing JJV into U'U'K Similarly CO 0^ 1 1 l2 2) becomes (12 . 13 . 23)* . UUU, ro t 0] 1 1 .2 2) becomes 12. 13.28 U'*V^^U\ ro 0] 1 1 2 2 3 V sj zr— _— . S becomes (12 . IS . 14 . 23 . 24 . 84) UUUU, &c The analogy would be closer if in the memoir on Linear Transformations, just as 12 is used to signify , 123 had been used to signify kic,, for then ro 0] 1 1 u 2) would have corresponded to 123 .UUU, ro 0] 1 1 u 2} to 128 £r-»[7'I7»; and this would not only have been an addition of some importance to the theoij*, but would in some instances have facilitated the calculation of hyperdeterminants. The preceding remarks show that the intermutant ro 0^ 1 1 T U 1 ^J (where the first and fourth blanks in the last column are to be considered as belonging to the same set) is in the hyperdeterminant notation (12 . 34)*.(14. 23) [7Z7[7£r. 1 Viz. corresponds to l2 beoaase and 1 are the characters ooonpying the first and second blanks of a oolomn. 1 If and 1 had been the characters occapying the second and third blanks in a column, the symbol would have been 23 and so on. It will be remembered, that the symbolic nombers 1, 2 in the hyperdeterminant notation are merely introdaoed to distinguish from each other functions which are made identical after certain differentiations are performed. 104] ON THE THEORY OF PERMUTANTS. ?3 It will, I think, illustrate the general theory to perform the development of the last-mentioned intermntant. We have '0 0^ as + 0' — t 0' — t '0 r + t ^0 1^ 111 1 I 1 110 1 1 1 1 1 1 1 1 ll 1 IJ a 1 1. .111- A 1 0. .1 1 0> ^2 (TO on ro oi-ro o] ro o it iLi 1 iJ Ll 1 iJ Ll 1 oj Ll 1 iJ = 2{(ad-6c)«-4(ac-6»)(W-c»)}, = 2 (a»(? + 4ac» + 46»rf - 36»c» - 6abcd), the different steps of which may he easily verified. The following important theorem (which is, I believe, the same as a theorem of Mr Sylvester's, published in the Philosophical Magazine) is perhaps best exhibited by means of a simple example. Consider the intermutant 'x r y 4 X 3 u 2. where in the first column the sets are distinguished as before by the horizontal bar, but in the second column the 1, 2 are to be considered as forming a set, and the 3, 4 as forming a second set. Then, partially expanding, the intermutant is 'x r — 'y V — 'x V + 'y r y 4 X 4 y 4 X 4 X 3 X 3 y 3 y 3 y 2. y 2. <x 2> .X 2. or, since entire horizontal lines may obviously be permuted, + 'x V — t 'y r •^ + 'x I' + t > 1^ y 2 y 2 X 2 X 2 X 3 X 3 y 3 y 3 ^X 4; .y 4> .« 4> .y *. 24 ON THE THEORY OF PERMUTANTS. [104 and, observing that the 1, 2 form a permutable system as do also the 3, 4, the second and third terms vanish, while the first and fourth terms are equivalent to each other; we may therefore write 'x V s 'x V y 2 y 4 X 3 X 3 .y 4> .y 2. where on the first side of the equation the bar has been introduced into the second column, in order to show that throughout the equation the 1, 2 and the 3, 4 are to be considered as forming distinct sets. Consider in like manner the expression (x U y z 7 6 X 8 y 2 z 9 X 4 y 5 ^z Sj where in the first column the sets are distinguished by the horizontal bars and in the second column the characters 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be considered as belonging to distinct sets. The same reasoning as in the former case will show that this is a multiple of X V y 2 z 3 X "i y 5 z 6 X 7 y 8 J 9. and to find the numerical multiplier it is only necessary to inquire in how many wajTS, in the expression first written down, the characters of the first column can be 104] ON THB THEORY OF PBRMUTANTS. 25 permuted so that x, y, z may go with 1, 2, 3 and with 4s, 5, 6 and with 7, 8, 9. The order of the x, y, z ia the second triad may be considered as arbitrary; but onoe assumed, it determines the place of one of the letters in the first triad; for instance, xS and z9 determine y7. The first triad must therefore contain xl and z6 or x6 and zl. Suppose the former, then the third triad must contain zS, but the remaining two combinations may be either x4i, y5, or x5, y4. Similarly, if the first triad contained x6, zl, there would be two forms of the third triad, or a given form of the second triad gives four different forms. There are therefore in all 24 forms, or t 24 'x V = 'x r y 2 y 7 z 3 X 4 z 6 X 8 y 5 y 2 z 6 X 7 z 9 X 4 y 8 y 5 .-^ 9. z 3 where the bars in the second column on the first side show that throughout the equation 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be considered as forming distinct sets. The above proof is in reality perfectly general, and it seems hardly necessary to render it so in terms. To perceive the significance of the above equation it should be noticed that the first side is a product of determinants, viz. 24 'x r t 'x 6^ t 'x 7' y 2 y 6 y 8 .z 3. .z 7. .z 9. t; and if the second side be partially expanded by permuting the characters of the second column, each of the terms so obtained is in like manner a product of deter- minants, so that 24 / X r t 'x 4^ t 'x T t = 'x V t 'x 8^ t 'x 4^ y 2 y 5 y 8 y 7 y 2 y 5 .e 3. .e 6. .» 9> .z 6. .« 9. .« 3. + ±&c., the permutations on the second side being the permutations inter se of 1, 2, 3, of 4, 5, 6, and of 7, 8, 9. It is obvious that the preceding theorem is not confined to intermutants of two columns. c. n. 4 26 ON THE THEORY OF PBRMUTANTS. [104 POSTSCRIPT. I wish to explain as accurately as I am able, the extent of my obligations to Mr Sylvester in respect of the subject of the present memoir. The term permutant is due to him — ^intemiutant and commutant are merely terms framed between us in analogy with permutant, and the names date from the present year (1851). The theory of commutants is given in my memoir in the Cambridge PhUo- iophical Transactionij [12], and is presupposed in the memoir on Linear Transformations, [13, 14]. It will appear by the last-mentioned memoir that it was by representing the coefficients of a biquadratic function by a = 1111, 6 = 1112 = 1121 «=&c, c=1122s&c., c?=1222 = &c., 6 = 2222, and forming the commutant Mlll^ that I was led to the function (w-4W+3c*. The function aoe+ibcd-cuP-h-e-i"^ [ 2222 J or a, 6, c is mentioned in the memoir on Linear Transformations, as brought into notice by bf Cj d c, dy e Mr Boole. From the particular mode in which the coefficients a, 6,... were represented by symUdH such as 1111, &c., 1 did not perceive that the last-mentioned function could be expressed in the commutant notation. The notion of a permutant, in its most general sense, is explained by me in my memoir, "Sur les determinants gauches,'' Ordley t xzzvii. pp. 93 — 96, [69]; see the paragraph (p. 94) commencing " On obtient ces fonctions, &c,** and which should run as follows : *' On obtieiit oes fonctions (dont je reprends ici la thdorie) par les propriety gdnerales d'un determinant d^fini comme suit. En exprimant &c. ;" the sentence as printed being " d^fini. Car en exprimant &c.,'' which confuses the sense. [The paragraph is printed correctly 69, p. 411.] Some time in the presieut year (1851) Mr Sylvester, in conversation, made to me the very important remark, that as one of a class the above-mentioned function, aoe + ibcd -ad^^l^-c^, could be expressed in the commutant notation ( ^ , viz. by considering 00 = a, 01 «» 10 = />, 1 1 .2 2; 02 = llai20=c, 12»21=cf, 22 =e; and the subject being thereby recalled to my notice, I found shortly afterwards the expression for the function a W + 4ac» + 463rf - 36M - Qabod (which cannot be expressed as a commutant) in the form of an intermutant, and 1 was thence leii to see the identity, so to say, of the theory of hyperdeterminants, as given in the memoir on Linear Transformations, with the present theory of intermutants. It is understood between Mr Sylvester and myself, that the publication of the present memoir is not to affect Mr Sylvester's right to claim the origination, and to be considered as the first publisher of such part as may belong to him of the theory hero sketched out. 105] 27 105. CORRECTION OF THE POSTSCRIPT TO THE PAPER ON PERMUTANTS. [From the Cambridge and Dublin MathenuUical Journal, vol. vii. (1852), pp. 97 — 98.] Mr Sylvester has represented to me that the paragraph relating to his com- munications conveys an erroneous idea of the nature, purport, and extent of such communications; I have, in fact, in the paragraph in question, singled out what imme- diately suggested to me the expression of the function Qabcd + 36*0* — 400* — 46*^ — a'd* as a partial commutant or intermutant, but I agree that a fuller reference ought to have been made to Mr Sylvester's results, and that the paragraph in question would more properly have stood as follows: '* Under these circumstances Mr Sylvester communicated to me a series of formal statements, not only oral but in writing, to the effect that he had discovered a permutation method of obtaining SLA many invariants — viz. commutantive invariants — by direct inspection from a function of any degree of any nimiber of letters as the index of the degree contains even factors ; one of these conunu- tantive invariants being in fact the function ace-^-^hcd-ae^-bd^-i^^ expressible, according to Mr fa^ ah 6*\ Sylvester's notation, l>y ( 2' ^ jsl)'* ^^d, according to the notation of my memoir in the Camb. PhU. Tram., supposing 00= a, 01 = 10=6, 02 = ll = 20=c, &c. by 00 11 22 » Mr Sylvester and I shall, I have no doubt, be able to agree to a joint statement i>f any further correction or explanation which may be required. 4—2 28 [106 106. ON THE SINGULAKITIES OF SUEFACES. [From the Cambridge and Dublin Mathematical Journal, vol. vii. (1852), pp. 166 — 171.] In the following paper, for symmetry of nomenclature and in order to avoid ambiguities, I shall, with reference to plane curves and in various phrases and compoimd words, use the term •*node" as synonymous with double point, and the term "spinode" as synonymous with cusp. I shall, besides, have occasion to consider the several singularities which I call the "flecnode," the "oscnode," the "fleflecnode," and the "tacnode:" the flecnode is a double point which is a point of inflexion on one of the branches through it; the oscnode is a double point which is a point of osculation on one of the branches through it; the fleflecnode is a double point which is a point of inflexion on each of the branches through it; and the tacnode is a double point where two branches touch. And it may be proper to remark here, that a tacnode may be considered as a point resulting from the coincidence and amalga- mation of two double points (and therefore equivalent to twelve points of inflexion); or, in a different point of view, [?] as a point uniting the characters of a spinode and a flecnode. I wish to call to mind here the definition of conjugate tangent lines of a surfiEu;e, viz. that a tangent to the curve of contact of the surface with any circumscribed developable and the corresponding generating line of the developable, are conjugate tangents of the surface. Suppose, now, that an absolutely arbitrary surface of any order be intersected by a plane: the curve of intersection has not in general any singularities other than such as occur in a perfectly arbitrary curve of the same order; but as a plane can be made to satisfy one, two, or three conditions, the curve may be made to acquire singularities which do not occur in such absolutely arbitrary curve. Let a single condition only be imposed on the plane. We may suppose that the curve of intersection has a node; the plane is then a tangent plane and the node is the point of contact — of course any point on the surface may be taken for 106] ON THE SINGULARITIES OF SURFACES. 29 the node. We may if we please use the term "nodes of a surface," "node-planes of a surface/' as synonymous with the points and tangent planes of a surface. And it will be convenient also to use the word node-tangents to denote the tangents to the curve of intersection at the node; it may be noticed here that the node-tangents are conjugate tangents of the surface. Next let two conditions be imposed upon the plane: there are three distinct cases to be considered. First, the curve of intersection may have a flecnode. The plane (which is of course still a tangent plane at the flecnode) may be termed a flecnode-plane ; the flecnodes are singular points on the surface lying on a curve which may be termed the " flecnode-curve ^" and the flecnode-planes give rise to a developable which may be termed the flecnode-develope. The " flecnode-tangents " are the tangents to the curve of intersection at the flecnode; the tangent to the inflected branch may be termed the "singular flecnode-tangent," and the tangent to the other branch the "ordinary flecnode-tangent." Secondly, the curve of intersection may have a spinode. The plane (which is of course still a tangent plane at the spinode) may be termed a spinode-plane ; the spinodes are singular points on the surface lying on a curve which may be termed the "spinode-curve*." And the spinode-planes give rise to a developable which may be termed the " spinode-develope." Also the " spinode-tangent " is the tangent to the curve of intersection at the spinode. Thirdly, the curve of intersection may have two nodes, or what may be termed a "node-couple." The plane (which is a tangent plane at each of the nodes and therefore a double tangent plane) may be also termed a "node-couple-plane." The node-couples are pairs of singular points on the surface lying in a curve which may be termed the "node-couple-curve," and the node-couple-planes give rise to a deve- lopable which may be termed the " node-couple-develope." The tangents to the curve of intersection at the two nodes of a node-couple might, if the term were required, be termed the "node-couple-tangents." Also one of the nodes of a node-couple may be termed a " node-with-node," and the tangents to the curve of intersection at such point will be the " node-with-node-tangents." 1 The fleonode-onrve, defined as the looas of the points through which can be drawn a line meeting the surface in four conseoaftive points, was, so far as I am aware, first noticed in Mr Salmon's paper **0n the Triple Tangent Planet of a Surface of the Third Order'* {Journal, t. iv. [1849], pp. 252—260), where Mr Salmon, unong other things, shows that the order of the surface being fi, the curve in question is the intersection of the surface with a surface of the order lln-24. * The notion of a spinode, considered as the point where the indicatriz is a parabola (on which account the spinode has been termed a parabolic point) may be found in Dupin's Diveloppements de OiomStrie : the most important step in the theory of these points is contained in Hesse's memoir "Ueber die Wendepuncte der Curren dritter Ordnnng" {CreUe, t. xxvin. [1848], pp. 97 — 107), where it is shown that the spinode-curve is the curre of intersection of the surface supposed as before of the order n, with a certain surface of the order 4(fi-2). See also Mr Salmon's memoir "On the Condition that a Plane should touch a surface along a Corre Line" {Jovrnal, t. in. [1848], pp. 44—46). 30 ON THE SINGULARITIES OF SURFACES. [l06 It is hardly necessary to remark that the flecnode-curve is not the edge of regression of the fleenode-develope, and the like remark applies m,fn. to the spinode- curve and the node-couple curve. Finally, let three conditions be imposed upon the plane: there are six distinct cases to be considered, in each of which we have no longer curves and developes, but only singular points and singular tangent planes determinate in number. First, the curve of intersection may have an oscnode. The plane (which continues a tangent plane at the oscnode) is an " oscnede-plane.'' The '' oscnode-tangents " are the tangents to the curve of intersection at the oscnode ; the tangent to the osculating branch is the " singular oscnode-tangent ; " and the tangent to the other branch the "ordinary oscnode-tangent." Secondly, the curve of intersection may have a fleflecnode. The plane (which continues a tangent plane at the fleflecnode) is a " fleflecnode-plane." The " fleflec- node-tangents " are the tangents to the curve of intersection at the fleflecnode. Thirdly, the curve of intersection may have a tacnode. The plane (which continues a tangent plane at the tacnode) is a " tacnode-plane.'' The ''tacnode- tangent" is the tangent to the curve of intersection at the tacnode. Fourthly, the curve of intersection may have a node and a flecnode, or what may be termed a node-and-flecnode. The plane (which is a tangent plane at the node and also at the flecnode, where it is obviously a flecnode-plane) is a "node-and- flecnode-plane." The " node-and-flecnode-tangents," if the term were required, would be the tangents to the curve of intersection at the node and at the flecnode of the uode-and-flecnode. The node of the node-and-flecnode may be distinguished as the node-with-flecnode, and the flecnode as the flecnode-with-node, and we have thus the terms " node-with-flecnode-tangents," " flecnode-with-node-tangents," " singular flecnode- with-node-tangent," and "ordinary flecnode- with-node-tangent." Fifthly, the curve of intersection may have a node and also a spinode, or what may be termed a " node-and-spinode." The plane (which is a tangent plane at the node, and is also a tangent plane at the spinode, where it is obviously a spinode-plane) is a " node-and-spinode-plane." The node-and-spinode-tangents, if the term were required, would be the tangents at the node and the tangent at the spinode of the node-and-spinode to the curve of intersection. The node of the node-and-spinode may be distinguished as the " nodo-with-spinode," and the spinode as the "spinode- Avith-node," and we have thus the terms " node-with-spinode-tangent," " spinode-with-node- tangent." Sixthly, the curve of intersection may have three nodes, or what may be termed a "node-triplet." The plane (which is a triple tangent plane touching the surfieu^ at each of the nodes) is a "node-triplet-plane." The "node-triplet-tangents," if the term were required, would be the tangents to the curve of intersection at the nodes of the node-triplet. Each node of the node-triplet may be distinguished as a "node- 106 J ON THE SINGULARITIES OP SURFACES. 31 with-node-couple," and the tangents to the curve of intersection at such nodes are •'node-with-node-couple-tangents." The terms " node-couple-with-node," '' node-couple-with- node-tangent," might be made use of if necessary. It should be remarked that the oscnodes lie on the flecnode-curve, as do also the fleflecnodes; these latter points are real double points of the flecnode-curve. The tacnodes are points of intersection and (what will appear in the sequel) points of contact of the flecnode-curve, the spinode-curve, and the node-couple-curve. The spinode- with-nodes are points of intersection of the spinode-curve and node-couple-curve, and the flecnode-with-nodes are points of intersection of the flecnode-curve and node-couple- curve; the node-with-node-couples are real double points (entering in triplets) of the node-oouple-curve. Consider for a moment an arbitrary curve on the surface; the locus of the node- tangents at the different points of this curve is in general a skew surface, which may however, in cases to be presently considered, degenerate in different ways. Reverting now to the flecnode-curve, it may be shown that the singular flecnode- tangent coincides with the tangent of the flecnode-curve. For consider on a surface two consecutive points such that the line joining them meets the surfistce in two points consecutive to the first-mentioned two points. The line meets the surface in four consecutive points, it is therefore a singular flecnode-tangent ; each of the first- mentioned two points must be on the flecnode-curve, or the singular flecnode-tangent touches the flecnode-curve. The two flecnode-tangents are by a preceding observation conjugate tangents. It follows that the skew surface, locus of the flecnode-tangents, brides up into two surfaces, each of which is a developable, viz. the locus of the singular flecnode-tangents is the developable having the flecnode-curve for its edge of regression, and the locus of the ordinary flecnode-tangents is the flecnode-develope. Of course at the tacnode, the tacnode-tangent touches the flecnode-curve. Passing next to the spinode-curve, the spinode-plane and the tangent-plane at a consecutive point along the spinode-tangent are identical^ or their line of intersection is indeterminate. The spinode-tangent is therefore the conjugate tangent to any other tangent line at the spinode, and therefore to the tangent to the spinode-curve. It follows that the surfEtce locus of the spinode-tangents degenerates into a developable sar£BU» twice repeated, viz. the spinode-develope. Consider the tacnode as two coin- cident nodes; each of these nodes, by virtue of its constituting, in conjunction with the other, a tacnode, is on the spinode-curve; or, in other words, the tacnode-tangent touches the spinode-curve, and the same reasoning proves that it touches the node- couple-curva It has already been seen that the tacnode-tangent touches the flecnode- curve ; consequently the tacnode is a point, not of simple intersection only, but of omtact, of the flecnode-curve, the spinode-curve, and the node-couple-curve. In virtue of the principle of the spinode-plane being identical with the tangent plane at a consecutive point along the spinode tangent, it appears that the tacnode- 1 It most not be inferred that the tangent plane at rach oonBeeotiTe point is a spmode-plane ; thia ii obfTioody not the ease. 32 ON THE SINGULARITIES OF SURFACES. [106 plane is a stationary plane, as well of the flecnode-develope as of the spinode- develope, and it would at first sight appear that it must be also a stationary tangent plane of the node-couple-develope. But this is not so; the node-with-node- planes envelope, not the node-couple-develope, but the node-couple-develope twice repeated: the tacnode-plane is in a sense a stationary plane on such duplicate developable, but not in any manner on the single developable. The tacnode-plane is an ordinary tangent plane of the node-couple-develope. Consider now a spinode-with-node, which we have seen is a point of intersection of the spinode-curve and node-couple-curve. The tangent plane at a consecutive point along the spinode-with-node-tangent, is identical with the spinode-with-node-plane ; the curve of intersection of the tangent plane at such consecutive point has therefore a node at the node-with-spinode, or the tangent plane in question is a node-couple- plane, and the point of contact is a point on the node-couple-curve. Consequently the spinode-with-node-tangent touches the node-couple-curve, and thence also the spinode-with-node-plane is a stationary tangent plane of the node-couple-develope. It should be remarked that no circumscribed developable can have a stationary tangent plane except the tangent planes at the points where the curve of contact meets the spinode-curve, and any one of these planes is only a stationary plane when the curve of contact touches the spinode-tangent ; and that the node-couple- curve and the flecnode-curve do not intersect the spinode-curve except in the points which have been discussed Recapitulating, the node-couple-curve and the spinode-curve touch at the tacnodes, and intersect at the spinode-with-nodes : moreover, the tacnode-planes are stationary planes of the spinode-develope, and the spinode-with-node-planes are stationary planes of the node-couple-develope. Besides this, the two curves are touched at the tacnodes by the flecnode-curve, and the tacnode-planes are stationary planes of the flecnode- develope. 107] 33 107. ON THE THEORY OF SKEW SURFACES. [From the Cambridge and Dublin Mathematical Journal, vol. vii. (1862), pp. 171 — 173.] A SURFACE of the n^ order is a surface which is met by an indeterminate line in n points. It follovrs immediately that a surface of the n^ order is met by an indeterminate plane in a curve of the n^ order. Consider a skew sur&ce or the surface generated by a singly infinite series of lines, and let the surfeice be of the n^ order. Any plane through a generating line meets the sur£Gtce in the line itself and in a curve of the (n — 1)^ order. The generating line meets this curve in (n — 1) points. Of these points one, viz. that adjacent to the intersection of the plane with the consecutive generating line, is a unique point ; the other (n — 2) points form a systenL Each of the (n — 1) points are svb modo points of contact of the plane with the surface, but the proper point of contact is the unique point adjacent to the intersection of the plane with the consecutive generating line. Thus every plane through a generating line is an ordinary tangent plane, the point of contact being a point on the generating line. It is not necessary for the present purpose, but I may stop for a moment to refer to the known theorems that the anharmonic ratio of any four tangent planes through the same generating line is equal to the anharmonic ratio of their points of contact, and that the locus of the normals to the sur&ce along a generating line is a hyperbolic paraboloid. Returning to the (n — 2) points in which, together with the point of contact, a generating line meets the curve of intersection of the sur&ce and a plane through the generating line, these are fixed points independent of the particular plane, and are the points in which the generating line is intersected by other generating lines. There is therefore on the surfSEU^ a double curve intersected in (n — 2) points by each generating line of the surfSsu^e — a property which, though insufficient to determine the order of this double curve, shows that the order cannot be less than (r - 2X (Thus for n = 4, the above reasoning shows that the double-curve must be an. 5 34 ON THE THEORY OF SKEW SURFACES. [l07 at least of the second order: assuming for a moment that it is in any case precisely of this order, it obviously cannot be a plane curve, and must therefore be two non- intersecting lines. This suggests at any rate the existence of a class of skew surfaces of the fourth order generated by a line which always passes through two fixed lines and by some other condition not yet ascertained; and it would appear that surfaces of the second order constitute a degenerate species belonging to the class in question.) In particular cases a generating line will be intersected by the consecutive generating line. Such a generating line touches the double curve. Consider now a point not on the surface ; the planes determined by this point and the generating lines of the surface are the tangent planes through the point; the intersections of consecutive tangent planes are the tangent lines through the point; and the cone generated by these tangent lines or enveloped by the tangent planes is the tangent cone corresponding to the point. This cone is of the n^ class. For considering a line through the point, this line meets the surface in n points, i.e. it meets n generating lines of the surface; and the planes through the line and these n generating lines, are of course tangent planes to the cone : that is, n tangent planes can be drawn to the cone through a given line passing through the vertex. The cone has not in general any lines of inflexion, or, what is the same thing, stationary tangent planes. For a stationary tangent plane would imply the inter- section of two consecutive generating lines of the surface. And since the number of generating lines intersected by a consecutive generating line, and therefore the number of planes through two consecutive generating lines, is finite, no such plane passes through an indeterminate point. The tangent cone will have in general a certain number of double tangent planes; let this number be x. We have therefore a cone of the class n, number of double tangent planes x, number of stationary tangent planes 0. Hence, if m be the order of the cone, a the number of its double lines, and fi the number of its cuspidal or stationary lines, 7M = n (n — 1) — 2a?, /3 = 3w (w - 2) - 6x, a = in (n - 2) (n» - 9) - 2« (w» - n - 6) + 2« (a: - 1). This is the proper tangent cone, but the cone through the double curve is sub modo a tangent cone, and enters as a square factor into the equation of the general tangent cone of the order n (n — 1). Hence, if X be the order of the double curve, and therefore of the cone through this curve, m -»- 2X = n (ti - 1), and therefore X = x\ that is, the number of double tangent planes to the tangent cone is equal to the order of the double curve. It does not appear that there is anything to determine x\ and if this is so, skew surfaces of the v!^ order may be considered as forming dififerent families according to the order of the double curve upon them. To complete the theory, it should be added that a plane intersects the surface in a curve of the n*** order having x double points but no cusps. 108] 35 108. ON CERTAIN MULTIPLE INTEGRALS CONNECTED WITH THE THEORY OF ATTRACTIONS. [From the Cambridge and Dublin Mathematicai Journal, vol. vii. (1852), pp. 174 — 178.] It is easy to deduce from Mr Boole's formula, given in my paper " On a Multiple Integral connected with the theory of Attractions," Journal, t. ii. [1847], pp. 219 — 223, [44], the equation df dv .^ f9-:it^ r «»-' {0x* -afds ^/{(-S(-$)■! where n is the number of variables of the multiple integral, and the condition of the integration is (g -«.)•, (1? - ft y , =,. also where and e is the positive root of , (g-g.)' . (8-$,y ^ir 0r jr + ^••■+7- € + — e+ — Suppose /= jr... = ^i, and write (a — o,)' + ... = i', we obtain ii(?-«)*+-«^]»"-« f{in-q)r(q+ljj . (l+«)*» • 5—2 36 ON CERTAIN MULTIPLE INTEGRALS [108 the limiting condition for the multiple integral being and the function a, and limit e, being given by € denoting, as before, the positive root. Observing that the quantity under the integral sign on the second side vanishes for « = e, there is no difficulty in deducing, by a differentiation with respect to Ou the formula i[(f-«)'...+t^]*^"T(in-g)r(g)j. where (2S is the element of the surface (^--^1)' + ... = ^i', and the integration is extended over the entire surfieu^e, A slight change of form is convenient We have if we suppose The formulae then become r df... TT** r (g|V -h x^ - i;')g (fa i[(f'-«)"- + «^F"^ r(in-9)r(9 + l)J. «(! + «)**+« • in which e is the positive root of the equation I propose to transform these formula by means of the theory of images ; it will be con- venient to investigate some preliminary formulae. Suppose X* = o" + /S" . . . , V = «i* + A*. . . ; also consider the new constants a, 6,..., Oi, 61,..., m, /i, determined by the equations where S is arbitrary. Then, putting 108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 37 it is easy to see that = V Proceeding to express the single integrals in terms of the new constants, we have in the first place A" = 8*4", where or if we write ooi + hhi ... = ZZi oos (k>, we have Hence also x~^3» where ,• - A* jfc. "' whence ._ 1 1 2%co8a> where p* = P + ii* — 2ZZi cos o), that is consequently ^jV + x* "" *^ = ^*n, where TI is given by n^ f' :^ iP'^^'-A') 8 — and it is clear that e will be the positive root of It may be noticed that, in the particular case of ti = 0, the roots of this equation are 0, and — — J}^ '^^. Consequently if f^—fi^ and li^-fi are of opposite signs, we have 6 = 0; but \t f-f^^ and ii'-/i» are of the same sign, €=^^ •'-7^--'— ^ 38 ON CERTAIN MULTIPLE INTEGRAX8 [108 In order to transform the double integrals, considering the new variables x, y, ..., I write 45* + y'... =r" and whence also, if f' + i7*+ ... =p* (which gives rp = S^), we have a: = — — . I • • * 9 also it is immediately seen tli*t (f-a)»+ ... «•= (ft ^^^.) ^ K^ - «)• + ••• +«*}. (f - a,)» ... -^,. = ^-^-^^{(a,-a,)»+ ... -/.'} : and from the latter equation it follows that the limiting condition for the first integral is (^— a,)'+... >/i' (there is no difficulty in seeing that the sign < in the former limiting condition gives rise here to the sign >), and that the second integral has to be extended over the surface (^ — 01)*+ ... =/i*. Also if dS represent the element of this surface, we may obtain did'n.,.^^dxdy ,.,, dX = ^^dS; and, combining the above formulae, we obtain r dxdy ... J (a^ + y« ... )*'»+« {(x - a)«-f (y- 6)' ... + i/'j*~-« "r(i^-9)r(j+i)(?+w«)*'»-^j. «(i+«)^+^' the limiting condition of the multiple integral being (^-a,)' + (y-6i)«...5/,«; and [ dS ~ r ( jn - 9) r? (i« + 1*«)*'^ (/i> -/i>) J . (1 + «)***^^* ' where d^f is the element of the surface (a?— ai)* + (y — Ji)" ... =/i', and the integration extends over the entire surface. In these formute, I, li, p, 11 denote as follows: p = a«+6»+..., /i' = ai» + 6i«+..., p» = (a-ai)> + (6-6x)»+ ... , and € is the positive root of the equation 11 = 0. 108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 39 The only obviously integrable case is that for which in the second fonnula q = l\ this gives / dS 2w^f, (a^ + y» ...)*~ {(a?- a)> + (y- 6)« + tA>}*'*-» r(iw)(P + t4»)^»(Zi»-/i')(l +€)*~-' ' In the case of t* = 0, we have, as before, when p*— /i* and l^—fi are of opposite signs, € = 0, and therefore 1 -f € = 1 ; but when p* — /i* and i,' — /i' are of the same sign, the value before found for 6 gives 1 + « = ^. i^/' + (p* -/>') (^' -/.*)}• Consider the image of the origin with respect to the sphere (a? — 01)*+ (y — 6i)"..^=/i', the coordinates of this image are > ••• » and consequently, if /a be the distance of this image from the point (a, 6 ...), we have /*'= {a- ^,(i.' -/.*)}»+.. . = ^ (^'/.' + (I'' -/x')W -/.')}; whence, by a simple reduction, or the values of the integral are /)•-/,» and k'-/i* opposite signs, ^ = f^^ F^^fc/O' n— 1 p* -/.' and /,' -/,• the same sign, / = ^?^^ ___^^__ , where fi is the distance from the point (a, 6...) of the image of the origin with respect to the sphere (x — a^y + . . . — /i' = 0. Stone Buildinga, AvguM 6, 1850. 40 [109 109. ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [From the Cambridge and Dublin Mathematical Journal, vol. viii. (1853), pp. 97 — 101] Suppose ^ + y = 0, ic» = a, y* = 6 ; then if we multiply the first equation by 1, j^, and reduce by the two others, we havt^ from which, eliminating x, y. x+ y = 0. bx + ay = 0, I, 1 = C b, a which is the equation between a and 6; or, considering x, y sa quadratic radicals, the rational equation between x, y. So if the original equation be multiplied by a-, //, we have a + xy-0, b + xy^O; or, eliminating 1, xy, a, 1 6, 1 = 0, which may be in like manner considered as the rational equation between x, y. The preceding results are of course self-evident, but by applying the same process to the equations ic + y + z = 0, a^ = ay y*^b, ^ = c, 109] ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. 41 we have results of some elegance. Multiply the equation first by 1, yz, zx, xy, reduce and eliminate the quantities x, y, z, xyz, we have the rational equation 111 =0; 1 . c 6 I c . a 1 6 a . and again, multiply the equation by x, y, z, xyz, reduce and eliminate the quantities 1, yz, zx, xy, the result is I a 6 c = 0, a . 1 1 6 1.1 ell. which is of course equivalent to the preceding one (the two determinants are in fact identical in value), but the form is essentially different. The former of the two forms is that given in my paper "On a theorem in the Geometry of Position" (Journal, voL II. [1841] p. 270 [1]): it was only very recently that I perceived that a similar process led to the latter of the two forma Similarly, if we have the equations X'\-y + z + w = 0, 5^ = a, ^" = 6, z^=^c, iv'^dy then multiplying by 1, yz, zXy xy, xw, yw, zw, an/zw, reducing and eliminating the quantities in the outside row. X. Vf '. tr, yzWf zwx, y>xy^ xyz we have the result 1 1 1 1 • • • • 1 1 1 c 6 c . a h a • • • 1 . . . 1 . . . 1 d . . . d . . . d a b c . 1 1 1 . 1 1 1 . • • • • • • • a b c d = 0; so if we multiply the equations by x, y, z, w, yzw, zwx, wxy, and xyz, reduce and eliminate the quantities in the outside row, C. II. 6 42 ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [109 1, yz, zXt xy^ xto, yw, zwt xytw we have the result a h c . 1 1 1 . 1 1 1 . 1 . . . 1 . . . 1 • • • d • • • 1 1 1 c b c . a b a •> 1 1 1 1 9 • • d . d . . . d • a b c • • • = 0, which however is not essentially distinct from the form before obtained, but may be derived from it by an interchange of lines and columns. And in general for any even number of quadratic radicals the two forms are not essentially distinct, but may be derived from each other by interchanging lines and columns, while for an odd number of quadratic radicals the two forms cannot be so derived from each other, but are essentially distinct. I was indebted to Mr Sylvester for the remark that the above process applies to radicals of a higher order than the second. To take the simplest case, suppose ^ + y = 0, a^-=a, y^=^b; and multiply first by 1, a^y, xy^\ this gives a? -f y . =0 bx + aj^y • = ; or, eliminating. 1 1 a 1 1 -0; next multiply by x, y, a^\ this gives x" or, eliminating. . -\-xy = y« + iry = 6a^ + ay* . =0; 1 . 1 =0; . 1 1 b a and lastly, multiply by aj*, y\ xy\ this gives b . + ary> = . ic*y + ay = ; 109] ON THE RATIONAUSATION OF CERTAIN ALQEBRAICAL EQUATIONS. 43 or, eUminating, a 1 . =0; 6 . 1 . 1 1 where it is to be remarked that the second and third forms are not essentially distinct, since the one may be derived from the other by the interchange of lines and columns. Applying the preceding process to the sjrstem multiply first by 1, asyz^ c^}^s^^ ^z, y'j?, ah/^ ah/, y*^, s^x, reduce and eliminate the quantities in the outside row, «. Vt «» y'«'. ^V't y***» '^^y* ***'» *V the result is ~^_ I I =0; 1 1 1 • • • • • • • • • 111 • • • • • • • • • a b c . a b . . . c . 1 . . . 1 . . . 1 . 1 . . . 1 1 . . . a . . . 6 c . 1 . . . 1 . . . 1 . . 1 1 . . . 1 . next multiply by a?, y, z, yV, s^a^^ a^y\ ahfz^ y*^a?, s^xy, reduce and eliminate the quantities in the outside row, the result is ""; I \ =0; 1 . . . 1 . . . 1 . 1 1 1 . 1 1 1 . • • • • • • • • • c 6 c . a 6 a • • • • • • • • • 1 . . . 1 . . . 1 • • • • • • • • • a . . 6 . . c . 1 1 1 . 1 1 1 . 6—2 44 ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [109 lastly, multiply by a^, y*, z\ yz, zx, xy, xt/^s^^ ys^^> xyV^ reduce and eliminate the quantities in the outside row, 1 xyz, zhjU\ yz\ «x», ary*, y*«, f'x, a?y the result is I" I = ; a b c • • • . 1 . . . 1 1 . . . . 1 1 . . . 1 • 1 1 1 1 . . . 1 . . . 1 1 . . . 1 . . . 1 • • • 1 1 1 c a . . b . . 6 . c a where, as in the case of two cubic radicals, two forms, viz. the first and third forms of the rational equation, are not essentially distinct, but may be derived from each other by interchanging lines and columns. And in general, whatever be the number of cubic radicals, two of the three forms are not essentially distinct, but may be derived from each other by interchanging lines and columns. 110] 45 110. NOTE ON THE TRANSFORMATION OF A TRIGONOMETRICAL EXPRESSION. [From the Cambridge and Dublin Mathematical Journal, vol. ix (1854), pp. 61 — 62.] The differential equation dx dy dz v = 0, (o + «) \/(c + a-") (a-^y)^{c + y) (a + z)>^(c + z) integrated so as to be satisfied when the variables are simultaneously infinite, gives by direct integration and, by Abel's theorem, 1, X, (a H- a?) V(c + a?) =0. 1, y, (a + y)V(c + y) , 1, z, (a + <2r) V(c + ^) ! To show d posteriori the equivalence of these two equations, I represent the deter- minant by the symbol Q, and expressing it in the form n = |lf a + x, (a + a:) V(c + a?) write for the moment f =. /f j &c. ; this gives 46 NOTE OK THE TRANSFORMATION OF A TRIGONOMETRICAL EXPRESSION. [llO n = 1. (a-c)(l + i). (o-c)»(^ + i) (o-c)'| f, f» + f p + 1 (o-c)* P. f f + 1 fVC» 1. f. f -f^? 1. e f } ^ (a-c )»(g+iy + g-fi?g) fvr 1. f f or, replacing (, i;, (f by their values, we have identically 1, X, (o + a;) \/(c + a?) 1, y, (p.+y)'J{c+y) 1, z, (o + «)V(c+z) (c+a;)*(c+y)'(c+g)* f Ajc, / «— c la—c ja—c ja-—^ /"~^l (a - c)* IV €+«"*■ V c+y V c+z V c+«V c+yV c+rj — c a— c c+a?' C+J? — c a— c /a— c c+y' c+y — c a— c C+-8:' C+r and the equation / g — c / g — c / g — c / g — c / g — c /g — c ^ ^ V c + a? V c + y V c + jgr Vc + a;V cTy V cT^ is of course equivalent to the trigonometrical equation tan- a/jHZ + tan-' a/^ + tan- a/^'' = 0. V c + « V c + y V c + £^ which shows the equivalence of the two equations in question. Ill] 47 111. ON A THEOREM OF K LEJEUNE-DIRICHLET'S. [From the Cambridge and Dublin Mathematical Journal, vol. ix. (1854), pp. 163 — 165.] The following formula, is given in Lejeune-Dirichlet's well-known memoir "Recherches sur diverses applica- tions &c." (Crelle, t. xxi. [1840] p. 8). The notation is as follows: — On the left-hand side (a, 6, c), (a', b\ c'), ... are a system of properly primitive forms to the negative determinant D (Le. a system of positive forms); x, y are positive or negative integers including zero, such that in the sum Sj*^"*'***^'^*''", aa^ + 2hxy 4- cy* is prime to 2jD, and similarly in the other sums ; q is indeterminate and the summations extend to the values first mentioned, of x and y. On the right-hand side we have to consider the form of jD, viz. we have D — PS' or else jD = 2PiSf, where S* is the greatest square factor in D and where P is odd: this obviously defines P, and the values of S, €, which are always ± 1 (or, as I prefer to express it, are always ±) are given as follows, viz. /) = PS*, P=l(mod4), S, €= + +, /> = PiS*, P = 3(mod4), S, € = -+, D = 2PiS*, P=l(mod4), S, € = + -, i> = 2P/S*, P = 3(mod4), S, € = , », n' are any positive numbers prime to 2Z), f pj is Legendre's symbol as generalized by Jacobi, viz. in general if /> be a positive or negative prime not a factor of n, 48 ON A THEOREM OF M. LEJEUNE DIRICHLET's. [Hl then ( - j = + or — according as n is or is not a quadratic residue of p (or, what is the same thing, p being positive, f— j = w*<**"« (modp)), and for P=pp'p*' .,., and the summation extends to all the values of n, nf of the form above mentioned. In the particular case jD = — 1, it is necessary that the second side should be doubled. The method of reducing the equation is indicated in the memoir. The following are a few particular cases. i) = - 1, X^"^^ = 42 (-)*<»»-« q^^\ 2) = — 2, V j«»+«y« = 22 ( — )* <**-" +* <***-*> g"*» , or (l + 25« + 25r«4-25r"...)(g + g* + g"4-g^ + ...) - 9 . _?•_ 2! £-_ + &c an example given in the memoir. i) = -3, 2?*"+'«^=22g)3**«', + 2(5'* + ^''^ J^ 4- J»*^4- ...)(3* + g"' + ?~ 4- ?>~ ...) ^g + g* g' + g* , g^ + g" g"-*-g " , l-V* 1-g" l-g« l-^"^*" I am not aware that the above theorem is quoted or referred to in any sub- sequent memoir on Elliptic Functions, or on the class of series to which it relates; and the theorem is so distinct in its origin and form from all other theorems relating to the same class of series, and, independently of the researches in which it originates, so remarkable as a result, that I have thought it desirable to give a detached state- ment of it in this paper. 112] 49 112. DEMONSTRATION OF A THEOREM RELATING TO THE PRODUCTS OF SUMS OF SQUARES. [From the Philosophical Magazine, vol. iv. (1852), pp. 515 — 519.] Mb Ktrkman, in his paper " On Fluquatemions and Homoid Products of Sums of n Squares" {Phil, Mag. vol. xxxiii. [1848] pp. 447 — 459 and 494 — 509), quotes from a note of mine the following passage: — "The complete test of the possibility of the pro- duct of 2* squares by 2** squares reducing itself to a sum of 2^ squares is the following : forming the complete sjrstems of triplets for (2^ — 1) things, if eab, ecd, fac, fdh be any four of them, we must have, paying attention to the signs alone, (±eab)(±ecd) = (±f(w)(± fdh) ; Le. if the first two are of the same sign, the last two must be so also, and vice versd; I believe that, for a system of seven, two conditions of this kind being satisfied would imply the satisfsu^tion of all the others : it remains to be shown that the complete system of conditions cannot be satisfied for fifteen thinga" I propose to explain the meaning of the theorem, and to establish the truth of it, without in any way assuming the exist- ence of imaginary units. The identity to be established is (ti;* + o« -h 6» + ...) (w; + a; + b; ...) = w,; + a,; + b,; + ... where the 2* quantities w, a, 6, c, ... and the 2** quantities w,, a„ 6„ c,, ... are given quan- tities in terms of which the 2^ quantities w,^, a,,, 5,,, c,„ ... have to be determined. Without attaching any meaning whatever to the symbols a^, 6^, c^ ... I write down the expressions w + aa^ + bb^ + cc^..., w, + afl^ + bf>^ + c,c^ . . . , an. 7 50 DEMONSTRATION OF A THEOREM RELATING [112 and I multiply as if a^, 6^, c^... really existed, taking care to multiply without making any transposition in the order inter se of two symbols a^, 6^ combined in the way of mul- tiplication. This gives a quasi-product tvw, + (aw, + a,w) a^ + (btv, + b,w) 6^ + • • • + aafl^* + bb,b^* + . . . -{-ab,aX + (^fibo<^o + "" Suppose, now, that a quasi-equation, such as means that in the expression of the quasi-product be, c a % d b , b , a c , b d are to be replaced by fl, 0, c, — ci,— 6, — c I *^o' o' o* o* o' o * and that a quasi-equation, such as ajb^c^ = — , means that in the expression of the quasi- product 6c. cd. a b , c b , ac. b (i are to be replaced by -^o» ""^o> "-c^» ^o> Ky Co- It is in the first place clear that the quasi-equation, ajbjc^ = +, may be written in any one of the six forms a6c=4-, 6c(i=4", ca6=4-, a c b =^ —, c6a=5:— , ((20= — ; ^0^0 o » *^o*'o o ' o**© o » and so for the quasi-equation ajb^c^ =» — . This being premised, if we form a system of (}uasi-equations, such as <^oMo = ±» ao^o«o = ±» &c- where the system of triplets contains each duad once, and once only, and the arbitrary signs are chosen at pleasure; if, moreover, in the expression of the quasi-product we replace a,', 6^', ... each by — 1, it is clear that the quasi-product will assume the form ^/y» ^y/» K* C// ••• b^iiig determinate functions of w, a, b, c, ...; w,, a,, b„ c, ..., homo geneous of the first order in the quantities of each set ; the value of w,, being obviously in every case w,^ = vm^ — tta, — 66, — cc, . . . , and a,,, b„, c,,,... containing in every case the terms aw,-{-a,w, btv, + b,w, cw,-¥ cjw,.,, but the form of the remaining terms depending as well on the triplets entering into the 112] TO THB PRODUCTS OF SUMS OF SQUARES. 51 system of quasi-equations as on the values given to the signs ± ; the qtiosi-egtuxtions serving, in fact^ to preset^ a rule for the formation of certain functions w,„ a^„ 6,,, c,,, ..., the properties of which functions may afterwards be investigated. Suppose, now, that the system of quasi-equations is such that e a b , e c d O O O' "O^O O being any two of its triplets, with a common symbol e^, there occur also in the system the triplets f fi c , f d b , Odd, a b c \ and suppose that the corresponding portion of the system is ^o^o^o = €, e^c^d^ = c', /o«oCo = (r, fodoK = ^» where €, f, *, e', ^, t each of them denote one of the signs + or — ; then e,„f„, g,, will contain respectively the terms € (ab, - a,b) + e' (cd, - c^d), i (ad^ — a,d) + if (be, — bfi) ; and e,,^+f*+g,* contains the terms (a> 4- 6» + c» + d«) (a; + b; + c/ + d;) - aJ'a; - H; - cFc^-dl'd; + 2 [ee {ab, - ajb) (cd, - c,d) + ^(ac,-afi){db,-'dfi) + u' {ad, - a,d) (be, - bfi)] ; and by taking account of the terms ew, + e,w, fw,+f,w, gw,-\-g,w in e„, f,„ gr^/ respect- ively, we should have had besides in ^^ -^-f^ + g^ ^® terms -\-2(ee,+ff^ + gg,)ww,. Also k;^' contains the terms ii/Hv,*-\'a*a,^ + b'b; + (^; + dM,^ -i(ee,+ff,+gg,)ww,] whence it is easy to see that (ti;* + a« + 6« + c*+ ...)(w/ + a/ + V + c/ + ...) + 2S [ee' (a&, - ajb) (cd, - c^d) + ^(ac,-a,c)(db,-d,b) + ii' (ad, — a,d) (be, — 6^c)]. 7—2 52 DEMONSTRATION OF A THEOREM RELATING &€. [112 where the summation extends to all the quadruplets formed each by the combination of two duads such as ab and cd, or ac and db, or ad and be, L e. two duads, which, com- bined with the same common letter (in the instances just mentioned e, or /, or g), enter as triplets into the sjrstem of quasi-equations — so that if i/ = 2** — 1, the number of quad- ruplets is i{H''-l)i(''-3)}v.i, = Av(.;- !)(,;- 3). and the terms under the sign 2 will vanish identically if only but the relation e^ = a' is of the same form as the equation cc' = ff ; hence if all the relations are satisfied, the terms under the sign S vanish, and we have {w^; 4- a^; + 6,/ + c,;+ ...) = («^ + a' + 6" + c«+ ...) (w; + a; + 6/ + c/ + ...) which is thus shown to be true, upon the suppositions — 1. That the sjrstem of quasi-equations is such that 6 db ^ 6 c d being any two of its triplets with a common sjrmbol e^, there occur also in the sjrstem the triplets 2. That for any two pairs of triplets, such as e a b , 6 c d and f a c , f d b , the product of the signs of the triplets of the first pair is equal to the product of the signs of the triplets of the second pair. In the case of fifteen things a, 6, o, ... the triplets may, as appears from Mr Kirk- man's paper, be chosen so as to satisfy the first condition; but the second condition involves, as Mr Kirkman has shown, a contradiction; and therefore the product of two sums, each of them of sixteen squares, is not a sum of sixteen squares. It is proper to remark, that this demonstration, although I think rendered clearer by the introduction of the idea of the system of triplets furnishing the rule for the formation of the expres- sions w,,, a,,, 6,„ c^,, &a, is not in principle different fi-om that contained in Prof. Young's paper "On an Extension of a Theorem of Euler, &c.", Irish Transactions, vol. xxi. [1848 pp. 311—341]. 113] 53 113. NOTE ON THE GEOMETRICAL REPRESENTATION OF THE INTEGRAL jdx^ >/(x + a) ~(aj + h){x + c). [From the Philosophical Magazine, voL v. (1863), pp. 281 — 284.] The equation of a conic passing through the points of intersection of the conies a^ + y* + ^ = 0, is of the form w (ic* + y* + 2:*) + cut" + 6y« + ag« = 0, where k; is an arbitrary parameter. Suppose that the conic touches a given line, we have for the determination oi w b, quadratic equation, the roots of which may be considered as parameters for determining the line in question. Let one of the values oi w he considered as equal to a constant quantity A;, the line is always a tangent to the conic *(«" + y* + ^) + flwj* + 6y' + C4^ = ; and taking w^p for the other value of t^;, /) is a parameter determining the parti- cular tangent, or, what is the same thing, determining the point of contact of this tangent The equation of the tangent is easily seen to be X ^h-c '/a + k VoTp + y'</c — a^b + k '^b+p-^-z 'Ja-h "Jc + k '^cTp^ ; suppose that the tangent meets the conic a^ + y^+z^ = (which is of course the conic corresponding to w = oo ) in the points P, P', and let d, oo be the parameters of the point P, and ^, oo the parameters of the point P', i.e. (repeating the defini- 54 NOTE ON THE GEOMETRICAL REPRESENTATION OF [ll3 tion of the terms) let the tangent at P of the conic a^H-y' + £^ = be also touched by the conic d(a^ + y' + ^') + aa?» + 6y» + 0^ = 0, and similarly for ff. The coordinates of the point P are given by the equations x:y:z=^ 'Jb-c Va + d: ^c-a 'JbTO : Va-6 Vc + tf ; and substituting these values in the equation of the line PP", we have (b - c) Va + A VoTp Va + d + (0-0) ^bTk y/b+p ^/W^ + (a - 6) \/c^k y/c+p Vc-f (? = 0. ..(♦), an equation connecting the quantities p, 0. To rationalize this equation, write V(a + i) (a +p)~(a + d)~= X + /ui, V(6 4-A)(6+i>)(6 + d) = X + ^, values which evidently satisfy the equation in question. Squaring these equations, we have equations from which X', \p^ fj} may be linearly determined ; and making the necessary reductions, we find X'* = a6c + kpd, - 2X/i= 6c + ca + a6-(p^4-Ap4- kd\ or, eliminating X, /[t, {6c + ca + a6-(p5 + ip4-A;d)}'-4(a4-6 + c + A?+p + d)(a6c + A:/>d) = 0, (♦), which is the rational form of the former equation marked (*). It is clear from the symmetry of the formula, that the same equation would have been obtained by the elimination of Z, M from the equations V (jfc + a) (A + 6) (Jfc + c) = Z + Mk, V(p + a)(/> + 6)(jp + c) ^L + Mp, and it follows from Abel's theorem (but the result may be verified by means of Euler's fundamental integral in the theory of elliptic functions), that if dx J • V(a! + o) (a + 6)(a! + c) ' then the algebraical equations (*) are equivalent to the transcendental equation ±m±nj)±nd=0; 113] THE i^n:m}RAh jdx-^J{x + a){x + b){x + c). 55 the arbitrary constant which should have formed the second side of the equation having been determined by observing that the algebraical equation gives for p^d, & = 00 , a system of values, which, when the signs are properly chosen, satisfy the transcendental equation. In fact, arranging the rational algebraical equation according to the powers of i, it becomes *»(p - ey - 2k {p^(p + ^) + 2 (a + 6 4-c)pe + (fcc + ca 4- ah){p + ^) + 2abc] -\'jfe'''2{bc-{-ca'\-ah)pd'' 4a6c (jp + d) + fr^c* + c»a» + a^6' - 2a^bc - 26«ca - 2c^ab = ; (♦) which proves the property in question, and is besides a very convenient form of the algebraical integral. The ambiguous signs in the transcendental integral are not of course arbitrary (indeed it has just been assumed that for p = 0y Up and RO are to be taken with opposite signs), but the discussion of the proper values to be given to the ambiguous signs would be at all events tedious, and must be passed over for the present. It is proper to remark, that d=p gives not only, as above supposed, &=x, but another value of k, which, however, corresponds to the transcendental equation ±Uk± 2np = ; the value in question is obviously , _ fi* - 2 (6c+ eg + ab)p^ - Sahcp + 6V + Ca' + a'6^ ~ 2a»6c - 2fe'co- 2(^06 "" (p + a) (p + 6) (|) + c) Consider, in general, a cubic function aai^ + Sba^y + 3ca:y* 4- dy*, or, as I now write it in the theory of invariants, (a, b, c, d) {x, yY, the Hessian of this function is (ac-b», Had-bc), bdr.c>)(a?, y)>, and applying this formula to the function (p 4- a) (p 4- 6) (p 4- c), it is easy to write the equation last preceding in the form lA'- D (a\b\c^ ^ Se8aiB,n {(p + a)(p 4- b)(p 4- c)} ^^~^ (a4-64-c) (p4-a)(p4-6)(p + c) which is a formula for the duplication of the transcendent Ilx, Reverting now to the general transcendental equation ±nA:±IIp±nd = 0, we have in like manner ±ni±np±n^=0; and assuming a proper correspondence of the signs, the elimination of Tip gives 56 NOTE ON THE GEOMETRICAL REPRESENTATION &C. [ll3 i.e. if the points P, P* upon the conic a;* + y' + 2r* = are such that their parameters 6, ff satisfy this equation, the line PP will be constantly a tangent to the conic k {x^ + y' 4- z") + {aa? + 6y« + c^) = 0. Hence also, if the paraibeters ^^ k\ Id' of the conies k (aJ» + y' + -2») + cw^ + 6y* + c^ = 0, k' (a^ + y' + -^*) + cur» + 6y»+C4:» = 0, A"(«* + y» + ^«) + (m;» + 6y> + c^» = 0, satisfy the equation nA: + nA' + nr = o, there are an infinity of triangles inscribed in the conic a;* + y' + -j* = 0, and the sides of which touch the last-mentioned three conies respectively. Suppose 2nA; = II/c (an equation the algebraic form of which has already been discussed), then = CO gives ff ^ K\ or, observing that ^ = oo corresponds to a point of intersection of the conies a^ + y" + £^ = 0, cw^ 4- 6y' + C2^ = 0, k is the parameter of the point in which a tangent to the conic A:(a;' + y* + -»®) + a^ + 6y' + c^ = at any one of its intersections ¥dth the conic aJ'-hy* + 4:* = meets the last-mentioned conic. Moreover, the algebraical relation between 0, ff and k (where, as before remarked, /v is a given function of k") is given by a preceding formula, and is simpler than that between d, ff and k. The preceding investigations were, it is hardly necessary to remark, suggested by a well-known memoir of the late illustrious Jacobi, and contain, I think, the extension which he remarks it would be interesting to make of the principles in such memoir to a system of two conies. I propose reverting to the subject in a memoir to be entitled "Researches on the Porism of the in- and circumscribed triangle." [This was, I think, never written.] 114] 57 114. ANALYTICAL RESEARCHES CONNECTED WITH STEINER'S EXTENSION OF MALFATTI'S PROBLEM. [From the Philosophical Transaciiona of the Royal Society of London, vol. CXLII. for the year 1852, pp. 253—278: Received April 12,— Read May 27, 1852] The problem, in a triangle to describe three circles each of them touching the two others and also two sides of the triangle, has been termed after the Italian geometer by whom it was proposed and solved, Malfatti's problem. The problem which I refer to as Steiner^s extension of Malfatti's problem is as follows: — "To determine three sections of a surface of the second order, each of them touching the two others, and also two of three given sections of the surface of the second order," a problem proposed in Steiner's memoir, "Einige geometrische Betrachtungen," CreUCy t. i. [1826 pp. 161 — 184]. The geometrical construction of the problem in question is readily deduced from that given in the memoir just mentioned for a somewhat less general problem, viz. that in which the surface of the second order is replaced by a sphere ; it is for the sake of the analytical developments to which the problem gives rise, that I propose to resume here the discussion of the problem. The following is an analysis of the present memoir: — § 1. Contains a lemma which appears to me to constitute the foundation of the aaalytical theory of the sections of a surfietce of the second order. § 2. Contains a statement of the geometrical construction of Steiner's extension of Mal£Eitti's problem. § 3. Is a verification, founded on a particular choice of coordinates, of the con- struction in question. § 4. In this section, referring the surface of the second order to absolutely general coordinates, and after an incidental solution of the problem to determine a section touching three given sections, I obtain the equations for the solution of Steiner's ^tension of Malfisitti's problem. an. 8 58 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 § 5. Contains a separate discussion of a sjrstem of equations, including a^ a particular case the equations obtained in the preceding section. ^ 6 and 7. Contain the application of the formulae for the general system to the eijuations in § 4, and the development and completion of the solution. § 8. Is an extension of some preceding formulae to quadratic functions of any number of variables. § 1. Lenwia relating to the sections of a surface of the second order. If flWJ" + fry* + c^ + dtu>' + 2fyz -h 2gzx + 2hxy + 2lxw -h 2mt/w -h 2nzw = be the equation of a surface of the second order, and the reciprocal equation, the condition that the two sections \'x -h fiy -h v'z -h p^w = 0, may touch, is (ax* -h iSfi^ + ©!/» + Bp' -h 2JpMi' + 2(Sv\ + i'f^XfjL + 2%\p -h i^fip -h 2^vp)^ X (av» + i8^'» + €v' + B/» + 2jffiV + 2(SvX + 2f^\v + 2'a\y -h 2iW/tv + 2^v'py + U(Xp' -h \'p)-hMW +M» + ia (vp'-^v'p)): and in particular if the equation of the surface be cux^ -\- bt/* -\- cz^ -\- 2fyz -h 2gzx -h 2hay-\-pvj^ = 0, the condition of contact is ax« +i8/i» +ffii/» +2JP/IAI/ -h2CBi/X+2|^X^-h-p»j (ax'« +i8M« +€^'' +2JF /iV + 2flri.V + 2?^XV + ^p in which last formula a=6c-/', 13*oa>-5r», aD=o6-A», § = gh-af , (S = 1if-bg. ^=fg-ch, K = ahc - of* - bf - ch* + 2/gh. 114] steimer's extension of malfatti's pboblem. 59 §2. In order to state in the most simple form the geometrical construction for the solution of Steiner's extension of Malfatti's problem, let the given sections be called for conciseness the determinators ^ ; any two of these sections lie in two different cones, the vertices of which determine with the line of intersection of the planes of the determinators, two planes which may be termed bisectors ; the six bisectors pass three and three through four straight lines ; and it will be convenient to use the term bisectors to denote, not the entire system, but any three bisectors passing through the same line. Consider three sections, which may be termed tactors, each of them touching a determinator and two bisectors, and three other sections (which may be termed separators) each of them passing through the point of contact of a determinator and tactor and touching the other two tactors ; the separators will intersect in a line which passes through the point of intersection of the determinators. The three required flections, or as I shall term them the resultors, are determined by the conditions that each resultor touches two determinators and two separators, the possibility of. the ooDstruction being implied as a theorem. The d posteriori verification may be obtained as follows: — § 3. Let J? = 0, y = 0, « = be the equations of the resultors, w; = the equation of the polar of the point of intersection of the resultors. Since the resultors touch two and two, the equation of the surfisice is easily seen to be of the form 2yz + 2zj? -h 2xy + k/« = 0. («) The determinators are sections each of them touching two resultors, but otherwise arbitrary; their equations are -ax + ^^y + ^z-^w^O, The separators are sections each of them touching two resultors at their point of contact (or what is the same thing, passing through the line of intersection of two resultors), and all of them having a line in common. Their equations may be taken to be cy — 6-2: = 0, cw^ — ca? = 0, 6a? - ay = 0, M use the words ** detenninatorB,'' <ftc. to denote indifferently the sections or the planes of the sections; the context is always saffident to prevent ambignity. * The reeiprooal form is, it should be noted, 8—2 60 ANALYTICAL RESEABCHES CONNECTED WITH [114 the values of a, 6, c remaining to be determined. Now before fixing the values of these quantities, we may find three sections each of them touching a detenninator at a point of intersection with the section which corresponds to it of the sections cy — 6-? = 0, aZ'-ca) = 0, for — ay = 0, and touching the other two of the last-mentioned sections ; and when a, 6, c have their proper values the sections so found are the tactors. For, let \x-\-fMy-\-vz + f>w = be the equation of a section touching the deter- minator — air-h^y-h^^ + t£; = 0, and the two sections &c — ay = 0, az — cx = 0: and suppose A« = X« + m' + »^ - 2/Av - 2i/X - 2X/I - 2p» ; the conditions of contact with the sections 6a? — ay = 0, ae — ca? = are found to be (6 4- a) A = (6 + a) \ - (6 + a) ft - (6 - a) I/, (c + a) A = (c + a) X — (c — a) fi - (c + a) I/, values, however, which suppose a correspondence in the signs of the radicals. Thence (b -\- a) fi = (c + a)v ; or since the ratios only of the quantities \ fi, v, p are material, fi = c '\' a, j^ = 6 + a, and therefore A« = X« - 2 (2a -h 6 4- c) X + (6 - cy - 2p«, = (X - 6 - cf, or p» = — 2 (aX + 6c). Hence the equation to a section touching 6a? — ay = 0, a^ — ca? = is Xj?4-(c + a)y + (6 + a)ir-hV -2(aX + 6c)\ w = ; and to express that this touches the detenninator in question, we have ± a (X - 6 - c) = (a + -) X - a (2a + 6 -h c) + 2 V-2(aX + 6c) ; and selecting the upper sign, 1 J - X-2aa = -2V-2(aX + 6c); whence X = - 2a {aa-^ " 26c), V - 2 (aX -h 6c) = (2aa - V ^6c) ; or the section touching the determinator and the sections 6a? — ay = 0, a-? — ca? = is -2a(aa- V-26c)«r-h(c + a)y + (6-ha)-? + (2aa->/-26c)w = 0; and at the point of contact with the determinator 2y« + &?« + 2ay + «;• = 0. 114] steiner's extension of malfatti's problem. 61 Eliminating w between the first and second equations and between the second and third equations, V-26c (ax -h - y 4- — ^^ + cy + 6-? = 0, and from these equations (cy — bzy = 0, or the point of contact lies in the section cy — bz^O. It follows that the equations of the tactors are -2a(aa-V-26c)a? + (c + a)y + (6 + a)-8:+(2aa- V' - 26c) w = 0, (c + 6) a: - 2/8 (6/8 - V - 2ca) y + (a + 6) -? + (26/8 - V - 2ca) w = 0, (6 + c) a: + (a + c) y - 27 (07 - V- 2a6) -? -h (2c7 - V-2a6) w = 0, where a, b, c still remain to be determined. Now the separators pass through the point of intersection of the determinators ; the equations of these give for the point in question, X : y : z : «; = (2/87 -h 1) (- a + /8 + 7 + 2a^7) (27a +1)( a-/8 + 7 + 2a/87) (2a/3 + l)( a4-/3-7 + 2a/37) 4a»/8V-l+a» + /9» + 7«; and the values of a, 6, c are therefore a :b : c = (2/87+ 1) (-a + i8 + 7 + 2a/37) :(27a+l)( a - /8 + 7 + 20/87) :(2a/8 + l)( a + /8-7 + 2a/87), which are to be substituted for a, 6, c in the equations of the separators and tactors respectively. Now proceeding to find the bisectors, let Tix + fiy + vz + pw^O be the equation of a section touching the determinators, ^«-/3y + ^* + w-0. ^x + ^y-yi + w = 0; and suppose, as before, A**X*-hfi* + i^ — 2/ij^ — 2i/X — 2X/i — 2p*; the conditions of con- tact are ±/8A = /3X-(/8 + i)/i + /8i/-2p, T 7A = 7X + 7/1 — f 7 + - j V — 2p, 62 ANALYTICAL RESEARCHES CONNECTED WITH [114 where it is necessary, for the present purpose, to give opposite signs to the radicals. For if the radicals had the same sign, it would follow that ^|^i8\-(/3 + ^)M-h/3i/--2pJ-^[^7X-h7/i-(7H-^),;-2pj=0; hence the section \x + fiy -{- vz -\- pw =^ would pass through the point nil 22 or the section would be a tangent section of the two determinators of the same class with the resultor a? = 0, which ought not to be the case. The proper formula is ^|^/3X-(^ + ~)AiH-/3i/-2p]+^[^7\ + 7/i-(7 + ^)i'-2p]=0; and this equation being satisfied, the section passes through a point «o 112 2 a: : y . z : w -- z : -gi^-y'-^"*- The bisector passes through this point and the line of intersection of the determi- nators ; its equation is l^x-fiy + lz + wy^{^x + ^y-yz + w)=0; or reducing and completing the system, the equations of the bisectors are In order to verify the geometrical construction, it only remains to show tliat each bisector touches two tactors. Consider the bisector and tactor -(^ + 2^)«+(l + 2^)y + (2^-2|')' + (-«-|)«' = ^' ■ - 2a (aa - V-26c) ar + (c + a)y-h(6-ha)-^ + (2aa- V-2ic) w = 0; and represent these for a moment by Xa? -h fiy + 1/-? + /Mi; = 0, X'a? + //y + p'z + p'w = ; 114] steiner's extension of malfatti's problem. 63 if A be the same as before, and A' the like function of \\ ja\ v\ p, also if * = XX' + /i/A' -h w' - (/lAi/' + fi'v) - (vV + v\) - (X/ + X» - 2pp', then -^■-(^^s)"' A'* = (200? - 2a V - 26c + 6 + c)», and the condition of contact AA' = <I> {taking the proper rign for the radicals) be- comes or reducing, "*-^^ + ''2^1 = ^' an equation which is evidently not altered by the interchange of a, a and 6, fi. The conditions, in order that each bisector may touch two tactors, reduce themselves to the three equations, -*'+*2-^i'^'^=<*' which are satisfied by the values dbove found £nr tk^ quantities a, h, c. The possi- hility and truth of the geometrical construction are thus demonstrated. Let it be in the first instance proposed to find the equation of a section touch in^^ all or any of the sections a? = 0, 3r = 0, £ = of the surfi^^ of the second order, ^wc" + 6y" + ci^ + Sfyz + 2gzx -h thxy + pw?'^ = 0. Any section whatever of this surface may be written in the form (aX + Afi + gv) x + (h\ + i/i -h /i/) y + (g\ -{-ffi + ci/) ^ + V — ;> V w = 0, where V« =:aX«+ V H-cr' + 2//iav + igvX -f 2A\/i - K, 64 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 and \ fi, V are indeterminate. And considering any other section represented by a like equation, {pX + hfi'-\-gi/) X -h (AX' -h bfi' +>') y + (flfV +// + cv') z + V^ V'w = 0, where V'« = aV» + 6/» + ci;'» + 2//j/' + 2fln^ V + 2A\y - K, it may be shown by means of the lemma previously given, that the condition of contact is aXX' + 6/a/a' + cvv' -\-f(p,v' + iLv) + g (i/X' + v\) + A (X/ia' + X» ± iT = VV. Suppose that X', y!^ v' satisfy the equations AV + 6/+yj/' = 0, gfX' +// + CI/' = 0, so that the last-mentioned section becomes a; = 0; and observing that the first of the above equations may be transformed into /» 1^ CBf it is easy to obtain X' = v W, /*' = "Tg , v' = j^ . The condition of contact thus becomes K and taking the under sign, X = a/^I, so that if in the above written equation we establish all or any of the equations X = V^, fi = V3B, v = V®, we have the equation of a section touching all or the corresponding sections of the sections «; = 0, y = 0, z = 0. In particular we have for a solution of the problem of tactions, the following equation of the section touching a? = 0, y = 0, jf = 0, viz. + — . Anticipating the use of a notation the value of which will subsequently appear, or putting f=^l^/vgB®-JF, g=-ys7vH3ffi; h = ^Vvi3B-|^, j=V24'gti3ar, 114] steiner's extension of malfatti's problem. 65 values which give ir« = -f*-g*-h* + 2g»h« + 2h«f« + 2f»g«-^^', the equation of the section in question is 7^<-f^ + g'-hh')a:-h;^(P-g'-hh')y-h^(P-hg'~h')^-h 'g^''/'''' i(; = 0. I proceed to investigate a transformation of the equation for the section with an indeterminate parameter X, which touches the two sections y = 0, z=^0. We have aV» = (oX + V + 9^y + (^/^^ + i8i/" - 2 jp/ii/) - J8ffi + Jp ; or putting for ^ and v their values VS, V(!D in the second term, aV'* = (aX + A/A + 5ri/)» + (VS® - Jp)» ; and introducing instead of X an indeterminate quantity X, such that aX + A/i + (7J^ = (VS®-jp)Z, we have also introducing throughout X instead of X, and completing the substitution of V33, v CD, for /A, V, the equation of the section touching y = 0, ^r = 0, becomes {aX'^hy'\'gz)X'k-y V® + -^ V:JB + w V -op Vl -h Z» = 0. It may be remarked here in passing, that this is a very convenient form for the demonstration of the theorem; "If two sections of a surface of the second order touch each other, and are also tangent sections (of the same class) to two fixed sections, then considering the planes through the axis of the fixed sections and the poles of the tangent sections, and also the tangent planes through this axis, the anharmonic ratio of the four planes is independent of the position of the moveable tangent sections;" where by the axis of the fixed sections is to be understood the line joining their poles. The sections touching -? = 0, a?=0, and iF = 0, y = 0, are of course x^-k-Qix + hy -\'fz) Y-\-z Va + w V-fcpVl-f r« = 0, a? VS +y Va + (flra; -h/y + c-?) Z + w "J ^cp Vl +i^''= 0, where Ax' +6/i' +/i;' =(V^a-<!B)7, X' = Vir / =/. »'' = V®, The conditions of contact of the sections represented by the above written equations would be perhaps most simply obtained directly from the lemma, but it is proper to deduce it firom the formula for contact used in the present memoir. If for shortness <I>(±) = aX'X" + 6/iV' -h cv V +/(/!/" -h Ai'V) + g {vX' + i/'V) + A (XV + X'V) ± K. C. II. 9 66 ANALYTICAL BBSEABCHES CONNECTED WITH [114 where the symbol 4>(±) is used in order to mark the essentially different character of the results corresponding to the different values of the ambiguous sign, then bap (_) =/(Ax' + V +>') (3^" +//*" + <»"). + 0SLv' - «SX' ) (g\" +/fi," + cp"). + (a/' - I^X") ihX' + 6/ +>' ). + v'fi" i-fSf)+ v'\"f^ + xy/ffi + XV (K -f$) = /(AX' + bfi,' +Jv') ig\" ^fy!' + cv") + Va (Va® - <S) O^X" +//*" + (»") + Va ( V^S - IQ) (AX' + hy! + >' ) +/C- a ^s®'+ 1^ v®if + ffi VI0B - a jF - («5?^ - a jF)) = /(AX' + 6/*' +/;') (jrX" +//*" + CI/") + v^ (Va® - ffi) («7X" +//' + CI/") + va (Vi® - 1^) (Ax' + 6/ + >' ) -/(VS® - (5) (Va23 - ?^), that is. 6c<I)(-) = (Va®-<ffi)(VaS-|^){/FZ + V'a(F+Z)-/l. What, however, is really required", is the value of 4>(+); to find this, we have 6c4> (+) = 6c^ (-) + 26cir =(V^(ff-(E)(vai8-i^)(/Fz+va(F+z)+/} + 26cir - 2/(Vaff - (K) (VaS - 1^), the second line of which is 2 (Va® - «i!) (v^is - 1^) 1^ (vn + ®) (vas + 1^) -/| ^2 (Va@-(SHVal8-i^) ^(^^^^^(^^^^)_^^^^^, = 2 (Va® - <!E)(>/ii9 - 1^) va^, 1 It may be shown without diffioolty that the (-) ngn would imply that the sections toaching 2 = 0, x=0, and £=0, y=0, were sections toaching ;c=0 at the same point. By taking the (-) sign in each equation we should have the solution of the problem **to determine three sections of a surface of the second order, the two sections of each pair touching one of three given sections at the same point," which is not without interest ; the solution may be completed without any difficulty. 114] steiner's extension of malfatti's problem. 67 where and consequently iK5<i>(+)==(V^(aD-ffi)(VaS-f^){/Fz+va(F+z)+/+2tfVa}, a reduction, which on account of its peculiarity, I have thought right to work out in full. The condition of contact is 4> (+) = VT'' = JL ( VS(^ - ffi) ( vai8 - 1^) vr+T^ ViTzi Hence finally, the condition in order that the sections (the former of which is a section touching £^ = 0, d; = 0, and the latter a section touching ir = 0, y = 0) may touch, is /Fz+va(F+z)+(/+2^va)-V6^vrTT^vnr^=o. The preceding researches show that the solution of Steiner's extension of Malfatti's problem depends on a system of equations, such as the system mentioned at the commencement of the following section. §5- Consider the sjmtem of equations a +i8 (F+z)+7 Yz j^h vnnr»vr+z» =0, «" + y8" (Z + F) + 7"ZF+ 8" Vr+X» VTTP = ; these equations may, it will be seen, be solved by quadratics only, when the coefficients satisfy the relations /3 yS' /3" 7-a y-a' V'-«"' ,y»_a» " 7'»-a^ 7"»-a"» 9—2 68 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 It may be remarked that these equations are satisfied by /8 = 0. /3' = 0, /3" = 0. 7 = 8, 7'=S', 7" = ^ or if we write a , o' a" - = — ^ -, = — m, -77 = — w, 7 7 7 the equations become by a simple reduction, 7»-hZ*+2i FZ = P -1, Z*-h Z«+ 2mZZ = m» - 1, Z»+P-h2nZF=n» -1, which are equivalent to the equations discussed in my paper " On a system of Equations connected with Malfatti's Problem and on another Algebraical System," Cambridge and Dublin Mathematical Joumat, t. iv. [1849] pp. 270 — 275, [79]; the solution might have been effected by the direct method, which I shall here adopt, of eliminating any one of the variables between the two equations into which it enters, and combining the result with the third equation. Writing the second and third equations under the form ^" + 5"z + cvr+z* = 0, the result of the elimination may be presented in the form which is most easily obtained by writing X=tan0 and operating with the symbol cos^^; but if the rationalized equations be represented by V + 2/Z + i;'Z» = and V' + 2/'Z + i/''Z» = 0, the form 4 {W - /i'«) {\%" - /'») = {W + \'V - 2//i'7 leads easily to the result in question. The values which enter are (7 = 8' VIT^, C" = S" VTTF'; whence, in the first place, by the equation connecting F, Z, C(7" = -?|l'{a + ^(F+Z)-h8FZ}. 114] steinbr's extension of malpatti's problem. 69 It is obviously convenient that A'A"-\-BB' should be symmetrical with respect to F and Z, and this will be the case if that is, if /8'(y'-a'0=/3"(7'-«'); or assuming that the equations are symmetrically related to the system, we have the first set of relations between the coefficients, relations which are satisfied by a = 7 + 20i8, a' = y + 20i8', a'' = 7" + 20/8", and the values of a, a\ a" will be considered henceforth as given by these conditions. We have il'^'' + 5'^'' - C^C^' = o'o'' + /8'/8'' + (7'/3'' + 7^/8' + 20/3'i8'O ( F + Z) + (^^^ H-^{a + ^(F-hZ)H-7FZ}. The quantities A'^-^-B^— C\ A"^ + B'^ — (7'» are quadratic functions of Z and Y respectively, and with proper relations between the coefficients, we may assume ^A'^'^B^'-C'^){A'''''^B'''^G'^)^W{V^^-kl{a^fi{Y^Z)^-r^YZy in which 17 is a linear function of Y '\- Z and FZ, and h and U are constants. The first side must, in the first place, be symmetrical with respect to F and Z, or must be proportional to But since («' + 7')/8'. (a" + 7")/9" are proportional to it is only necessary that should be proportional to y«-.a'«, 7"»-a"», ^a + yi-S'*, /3"« + y'«-S"» y«-a'», 7"*-a"»; or since the equations are supposed symmetrically related to the system, we must have the second set of relations between the coefficients. Suppose then since y-a» = -4(7 + 0/8)0)8, &c.. 70 ANALYTICAL RESEARCHES CONNECTED WITH [114 we have 8» «/8« +7» -*8(y +^fi)fi and S, S\ S"' will be supposed henceforth to satisfy these equations. We have next which may be simplified by writing u - <& l-\-ii4> g as t- ^ w :s 2-— where ^ v are to be considered as given functions of a and <f>. These values give A'* +R*-Cr* =4(7' +4>$')^s {Z + ,i){Z + v), ^"»+ £"«_ C"t = 4 (y + 0y3") ff'8{Y+it) (F+ 1/). Hence, putting for simplicity we have 4(Z + /*)(Z+i;)(F+At)(F+v)=£7» + *[(a+/3(F+Z) + 7F^-S'(l + F»)(l+Z')]; and the two sides have next to be expressed in terms of T + Z and YZ. If for symmetry we write f = l. 7, = 7+Z, K^YZ, then and U' is now to be considered a linear function of f, 17, 2^. The condition that the first side of the equation may divide into factors, gives an equation for determining k ; since the condition is satisfied for A; = and k-<x>, the equation will be linear, and it is easily seen that the value \a k^-^iji — vf. In fact hence {2,*vf + 0* + v) 1; + 2f }« - t7» = ^^i^ {(af + /3i, + 7f )• - S* (f + ?)■}. 114] steineb's extension of malfatti's problem. 71 and we may assume 2Mvf + 0* + .;)i7 + 2?-O-=^^{(af + ^i, + y?) + 8(f + ?)}, subject to its being shown that gives a constant value for A. The comparison of coefficients gives *~ a the first and third of these give {(A.l),-(A-i)4; 4 (1 - ^v) = ^g" (a + ^) (7 - «), which will be identical with the second, if 2(1 -Mv) ^ ^ ^ gA which follows at once from the equation Forming next the two equations ns 1/ = A + 1 a' 2 (M + •')«, "(m- -')/3 A- 1 A' "(m 2 -«'))3 Km + i')7- -2/9}, these will be equivalent to a single equation if (m + !/)• 8»= {0* + »')7 - 2/3{» + (/* - 1/)* /8», that is, if 0* + !/)• S» = 0* + 1/)" (/S* + 7») - 4 (/i + v) i87 - 4 (/*•; - 1) /S* ; 72 ANALYTICAL RESEARCHES CONNECTED WITH [114 or finally, if which is in fact the case. Writing ihe equations for A 1 A 1 in the form -^ + T = ;^ \"^-^"- x = 7 r5-(7- 2w, A (ji — vjps A {fi-vjPs^' and substituting in we have U='^\[^-^^(<'i+N+'iK)-[^+^»{^+K)\, ^ ((-/3 + 2«7 + 2^)f + (7-2«/3)i7 + (-/9+2»7 + 4«^/8)?}; and consequently, multiplying by we have or collecting the different terms which enter into the equation the result is - 1 V(7' + 2^/3')(7" + 20y8")/8'/8"l {(- /3 + 2*y + 2^) f +(7 - 2»^) 17 + (- /8 + 2*/ + 4«^/9)r} = 0. which, combined with the first equation written under the form (of + ^, + yO* - «• [(f - ry + 17»] = 0, determines the ratios of ^, 17, ^ that is, the values oi T + Z and YZ. 114] STiaNEB's EXTENSION OP MALPATTl's PROBLEM. 73 §6- The system of equations (/+ 20 V^) + V|l (F+ Z ) +/YZ - \/bc ^/T+Y' 'JT+Zi = 0. (g + 26 vIB) + VS(^ + X) +gZX - Vw \/l + Z* Vf+Z* = 0, (A + 2^ V^) + V® (Z + F) + AXF - V^ Vr+X« VlTT' = 0, where on which depends the solution of Steiner's extension of Malfatti's problem, is at once seen to belong to the class of equations treated of in the preceding section, and we have ^ = 0, « = 0. The equations at the conclusion of the preceding section become -a[(/+2^Vgr)f + Va, +/r] -^-J(g+e VS) (h+e V®) Vig® {('^-20f)^-fy + V|i?}=o. {(/+ 2^ va) I + v^, +/f}« - be {(f - (0* + »?'} = 0, which may also be written (VS® + Jf) (f + 5) + (- a Va + «7 V® + A ^18 + 25 Vl8®) (ij + 2^f ) - ^ -Jig + e VlJ) (A + 5 V®) Vfi®! ((VI - 25/) f -/i; + '/^d = 0, {/«+D + ^('? + 25f)}'-^Kf-0' + i?'} = 0. Hence observing that ^+5Vii = ^(Vi8® + JF)(Vai8 + |^); A + dV® = -^(VJ8®+iF)(Va® + ffi); -oVa+AVS+5'V®+25VS®=5(VS®+jp), and putting for a moment X = ^ ^/(V^ + (5) (Vaa8 + 1^) V3B© , and therefore V(^+ 5 VS)(A + 5 V«D) Vi8aD=(VJ8® + iF)x, a n. 10 74 ANALYTICAL RESEARCHES CONNECTED WITH [114 the first equation divides by (^^^8CD + Jp), and the result is Also, by an easy transformation, the second equation becomes or putting f+? + ^ (i? + 2^f ) = e, the equations become hence eliminating ^, e - 2x,o = 0, or observing that 1 + ^ = ;^. (^^® + iF) (v^^a + ®) (>^ai8 + ?^), and reducing, we obtain *=.^-=^ also B = 2X<I> gives Suppose then substituting V3iar+ jF = '*' V3S®- iF = «,. •■• «», = Ka, P/r, \ Va + a/ 114] STEINEE's extension of MALFATTfs PROBLEM. 75 that is, Pfli P.7/ V Va + a/ these may be written L'^ + if 17 + i\r'c= 0, where P,7/ V Vo+a/ or since f, ly, ? are equal to 1, T+Z, YZ respectively, 1: Y + Z: YZ= MIT - MN : NL' - ITL : LM - L'M ^_46c^)/ V2V^ A / _ ^U^J+3(/+,Va). ^-% ^ V/8,X >^ ^ Va + a/ V/8,7, Also whence y ^^^ _ 2V2V^Tv^/l _ V2^^^^N / _ V^N _ ^ ^ V)8,x A Va + a/ 2V2 V^T^, ^^ /^tf , ^2 Va + a/ \ / V;; N ^ V "^ v;^, Jr va+v ' and by forming the analogous expressions for Z-\-X and ZX, X+F and XY, the values of X, F, Z may be determined. But the equations in question simplify them- selves in a remarkable manner by the notation before alluded to. 10—2 76 ANALYTICAL RESEARCHES CONNECTED WITH [114 Suppose these values give iJ-^ =-f»-g'-h» + 2J>, K* =-f«-g'-h« + 2g»h» + 2h»f+2fV-^^'. Applying these results to the preceding formulae and forming for that purpose the equations » /s / la- A 1. ^2 Va + o" J* Vo, f 2 V2 Va + a, V^,7, = 4gh, 73 =-7s^. , — ^ = 7. V^,7/ Vagh Vo + a, «/ ghA-^ + ^ ITf = ( J« - gh) (f» - (g - h)*) - 2gh (g - hy. we have if(F+^ + 2Z^ = 4(J*-gh)(^l -^), iTTZ + Z« = {(J' - gh) (f» - (g - h)*) - 2gh (g - h)«} (1 - ^) ; the former of which, combined with the similar equations for Z-\-X and -T + F, gives for Z, F, Z the values to be presently stated, and these values will of course verify the second equation and the corresponding equations for ZX and XY. Recapitulating the preceding notation, if j? = 0, y = 0, z^O are the equations of the given sections, t(;=»0 the equation of the polar plane of their point of intersection with respect to the sur&ce, CM5" + 6y* 4- c-«* + 2^2? 4- 2gr^ + 2fcry +/)«;» = the equation of the sur&ce, ^, 38, ({D, $, <!Er, |^, iT as usual, and 114] steikeb's extension of malfatti's problem. 77 then the equations of the required sections are {ax + hy + gz)X'^y V® + ^ ^^ + w V - op V 1 + Z« = 0, X V® + y Va + (gx +/y + cz) Z+w^ - cpVl + Z* = 0, where X, F, ^ are to be determined by the following equations, (/+ 25>/a)+ va (Y+z)-hfYz - \/6c vrrF^viT^ = o, (y + 2^ VS) +vs(.^+z)+^^z-VcaV rr^ vi+z« = o, (A + 25 V®)+ V^(Z + F) + AZF- \/^ Vl+Z«Vrn^ = : and the solution of which, putting f=4'a>/vi5«^rS g=^©^/^^®a-ffi, h=^®>/vai3-?^, j^j~2<fmm. is given by the equations /irz = ^+( -.f+g + h)«-2(-f+g + h)j, ifF=?^ + ( f-g + h)>-2( f-g + h)J, ZZ = ?^^+( f + g-h)>-2( f+g-h)/. C) Instead of the direct but very tedious process by which these values of A", F, Z have been obtained, we may substitute the following d posteriori verification. We have K*(l+X*) =4(-f+g + h)'J'(l+^)(l-j)(l-J). 2f,VlTF»VrT^ = 4 (f -(jf-h )) /• (l - j) \/l - $y/l - J, • /r»(l + FZ) = * (l - j) {('^' -«*»> i^-(g-^y) - 2gh(g - h)",. ^(F+Z)-2f»- 2g«- 2h« + 4/» = 4 (l -^) (J'-gh). Putting also ^ It IB periiapB worth noticing that the value of the quantity X previously made use of, 78 we have ANALYTICAL RESEARCHES CONNECTED WITH [114 (f. _ g. _ h« + ?^') IT* (1 + FZ) = 4(l-J)|(p-(g-h)')[(J'-gh)(f«-(g-h)«)-2gh(g-h)«-2gh^=^^5^] 4g' h'(g-h)'(J»-g h) }■ hnj*-gh)\ J' K' {K(Y+Z) - 2f«- 2g'- 2h' + 4/'} = 4(i-;)j{f-(g-hy)[(j»-gh)((g+h).-f*-^?:)l-*«^- Also, since (f._(g.h).) + ((g+h).-f^-*^') = 4gh(:^>. we have (p -^ -h' +^^\ K' (1 + YZ) + K* {K (Y + Z) -it* -2^ -2h*} = 4(l-;J) (f«-(g-h).)2ghJ« (l -^.)(l- ^,). and the values obtained above give also 2gh \/l - ^\A-J. ^' ^ + ^' ^^1+^' =4(i-^)(f«-(g-h)02ghj«(i-f,)(i-^;). which shows that the relation between Y and Z is verified by the assumed values of these quantities, and the other two equations are of course also verified. The solution of the problem will be rendered more complete if the equations of the required sections and of the auxiliary sections made use of in the geometrical construction are expressed in terms of f, g, h, J. §7. First, to substitute in the equations of the required sections or resultors. the first equation in the form Writing K' the coeflBcient of x will be |aZa? + (AZ + V(!D)y + (grZ + V33)-^ + V-apVl + Z«w=0^ -^ ( ^-j^l^'+i-f+s+^y-^i'-f-^g-^^)]^ 114] steiner's extension of malfatti's problem. 79 or, as it is convenient to write it. 1 + ^)r(-'.».h,4(i-^){7p^^-f...h-^. The coeflBcient of y is ^{(-f.-g.^-.+^^X^'^*-'^*-'"^-"'-'*^^'')) 2Vi3 f 4 _ g4 _ h« + 2g»h' + 2hf + 2f g» - ^^^ or, after all reductions, i} 4)«-'-^-«^ (> -« {.-F^,*'-«-"-^m^h and similarly the coefficient of 2^ is ^|(-f.g--H-..^-e-)(?&t.(-,.,.h).-2^(-f.g^h,) _ f « _ g4 _ h« + 2g»h» + 2h»f + 2Pg' - ^^ j or, after all reductions. (,4)„.,.,..,^(,.h)|_^||^.,,,,.,,^2£^.b.,l and the coefficient of u; is (i + J)f(-f+g+h)2Vzyi-iy^i-&yi-^V-jt,. Hence, forming the equation of the resultor in question, and by means of it those of the other resultors, the equations of the resultors are , _2fgh_ U(-fVg + h) - f + g 4 h - 2/ )U-'^' + U(-f+g + h) + ^ g + h + 2J 2^^ j^\\ jjy I -2fgh .,, . orf*-g» + h»\ h /, h\ 80 ANALYTICAL RESEARCHES CONNECTED WITH [114 + 2 VF/^r^ /y/lTK y^T^ V^«, = values which might be somewhat simplified by writing |, t), ^, <o instead of U'-'i)'- U-i>' U'->- V^^/'^V'"^^- and it may be also remarked, that the coefficients as well of these formulae as of those which follow may be elegantly expressed in terms of the parts of a triangle having f, g, h for its sides. The equations of the separators are found by taking the differences two and two of the equations of the resultors (this requires to be verified d posteriori) \ thus sub- tracting the third equation firom the second the result contains a constant fector, j(f.-(g-h>')gh {^^g"^' -^ (f - (g - h)*) ((g+ by - p)}. equivalent to J(P-(g-h)^)ghV ^ V J^ J) (f«-(g-hy)gh Rejecting the factor in question and forming the analogous two equations, the equa- tions of the separators are ^ (aa' + ...:? S*) = (aa + f^/8 + (Syy, 114] steiner's extension of malfatti's pkoblem. 81 and from the mode of formation of these equations it is obvious that the separators have a line in common. The equations of the determinators being x = 0, y = 0, e^Q, the equations of the tactors are v'S«-V^y = 0, Vcf a; - V^« = 0, Vay- V3©a;=0; and if ax + fiy + yz + Bw = be the equation of the tactor touching 01 = 0, V®a!-v'a« = and Viiy-VSa! = 0, the conditions of contact are K P 2 V15B ( VIS - 1^) (aa" + . . . - 8*) = |( VISB - 1^) (a Va -/3 VS) + 7 (® VS - 1^ V^)l* , 2 VI® ( va® - ffi) (aa* + . . . - s*) = Ws® - (5) (a vfi - 7 V®) + ^ (?^ v® - jf va)l* . whence ;;^V2Va3B(VaS-|^)(aa + |^/3 + €J7) = (Vgoi - ^) vaa - (VIS - ^) vs/s + (ffi vs - jf va)7, ^V2Va®(va®-ffi)(aa+|^i8+ffi7) = (Va® - ffi) Vgfa + (1^ V® - Jf VI) /3 - (Vie - <!Ir) 7, and putting for a moment /i = vae - (5 - V2"viiaD(Val-€G), V = Vi® - 1^ - 72 vas (Vai8 - 1^); after some reductions, and observing that the ratios only of the quantities a, fi, y, S are material, we obtain K c. n. va" 11 82 ANALYTICAL RBSBARCHB8 CONNECTED WITH [ll4 and it is easily seen also that the coordinates of the point of contact are also « = 0, y = v, z^fi, t^ = - - -^- 5 '•- ^('-5). '=-^('-5)- Hence substituting and introducing throughout the quantities f, g, h, «/, also forming the analogous equations, the equations of the tactors are jp(-f.+g.+h.)+(g+h)j(f.-(g-h)--?^)j^. + 2>JK V gh (l - 5) (l - j) (f - (g-h)*) 'J~PW = 0, + 2V^ y hf (l - J) (l - j) (g" - (h - f )') -J"-^ = 0, + |h. (f. + g. - h«) + (f + g) / (h» - (f - g)» - ^ijj-*)! -^ ^ ^•JKsJigil - j) (l - f) (h'-(f-g)') ^^^^> = 0- + 114] steiner's extension op malfatti's problem. 83 It is obvious, from the equations, that each separator passes through the point of contact of a tactor and determinator, it consequently only remains to be shown that each separator touches two tactors. Consider the tactor which has been represented by aa; + /8y -f 7-8r 4- Sw = 0, the unreduced values of the coefficients give ^aa«+...|S» = ;;^(aa + f^5 + (E7) = iri Represent for a moment the separator h{^-7)'-U-?)y-'-^'U-^>"' hy lx + my ■hnz-h8W=: 0. Then putting ^P + ... — s» = Q', since = ^{,(i4)-,(i-^)(i-5)-(r-g)(i-^)(i-i)} the condition of contact becomes D=-r|-(f-g)'+h(f+g)-?^|; J or, forming the value of □' and subetituting, -.(.-^(r-,„(i_5)(,4)-.(,.^>,(,.5)(,_5) which may be verified without difficulty, and thus the construction for the resultors is shown to be true. 11—2 84 ANALYTICAL RESEARCHES CONNECTED WITH [114 §8 Several of the formulae of the preceding sections of this memoir apply to any number of variables. Consider the surface (Le. hypersurface) aai^ + bf + C2^+ 2fyz + 2gza) + 2hxy ... +pe* = 0, and the section (i.e. hypersection) where (aX + A/i +5fv ...) a? + {h\ + bii +fv ...) y + (ff^ +/a* H- ci' ••.) ^ ••• + ^ ""^ ^^ = 0, V* = a\* + bfi* + cv" + 2//X1/ H- 2gv\ + 2AV ... - K, the condition of contact with any other section represented by a similar equation is a\V + bfJLfi' + cw' +f(jiv + /jfp)-hg (vXf + v\) + h (Kf/ + \» ... ±K = W, where K is the determinant formed with the coeflScients a, 6, c, /, g, A, ... And con- sequently, by establishing all or any of the equations \ = \/^, /a = >/18, v = VCD, ... we have the condition in order that the section in question may touch all or the corresponding sections of the sections a? = 0, y = 0, z = Of ... Let tt be the number of the variables x, y, z... , then K^"^ =. G :ff (S^ also Z«-» {(a\ + hfM + gv ...) x + (h\ + bfi +fa ...) y + {9>^ -^ff^ + op ...) z ...] X y 2 X a 1^ ffi A* ?^ IS JF I* Ci iF ® whence also Z«-*(V» + ^ = - X /» V ... or /ir"-*v> = - X a 1^ e ^ ?^ 53 JF 1 " ffi iF 1 • ® 1 X /* V X ^ ?^ ® /* n 19 iF » €r iF € 114] STEINBBS EXTENSION OF MALFATTIS PBOBLEM. 85 and the equation of the section in question becomes X y z ... + ir*«->v-j>v- \ & n & t* n i3 JF V <& • • iF e 1 \ iF also the condition of contact with the corresponding section is Tl X /* V V » ?^ dr 1^ 13 iF l/ eEr iF ® V- 1 X /* V ... X a 1^ <!1t M n 13 iF • • • iF ® V ... iF ^ « = 0, V- 1 X' /*' y'... 1 X' a ^ dr /*' » as iF v' ® • • • iF e In particular the equation of the sections which touches all the sections x^O, y «=0, ..., is = 0, X y z V® <S jp CD + if jii-i V - p yn 1 Via vs v®... U=o. va a ?^ ffi v^ f^ 18 iF V^ (ffir iF ffi Again, the equations of the section touching y = 0, ^ = 0,... and the sections touching «sO, ^ = 0, ... are > = 0, tc y z X a ?^ dr v13 ?^ 33 iF • • • df iF e a; y 2 va a ?^ dr ^ ?^ 13 iF V® df iF ® +z»«-w-j3 7- 1 X V® • • • X Vi3 V®... a ?^ dr 1^ 55 iF di JF ® + Jfi«-W-p>/^ 1 V® • • • va" /* ^ - a ?^ dr 1^ 3D iF ffi JF ® 1< = o, 86 ANALYTICAL RESEABCHE8, &C. [114 and the condition of contact of these two sections is + 1 X Vi3 m sa 1^ /* m 13 V® & iF V<ZD... JF = V_" 1 X ^23 V®... \ ^ 1^ cs V39 1^ 38 iF V@ <!1t iF ® y- 1 va 18 iF iF ® It would seem from the appearance of these equations that there should be some simpler method of obtaining the solution than the method employed in the previous part of this memoir. 115] 87 115. NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. [From the Philosophical Magazine, voL VL (1853), pp. 99 — 102.] The equation of a conic passing through the points of intersection of the conies 17=0, F=0 is of the form wU+V=0, where i(; is an arbitrary parameter. Suppose that the conic touches a given line, we have for the determination o( w a, quadratic equation, the roots of which may be considered as parameters for determining the line in question. Let one of the values of w he considered as equal to a given constant k, the line is always a tangent to the conic kU+V = 0; and taking w^^p for the other value o{ w, p is a parameter determining the particular tangent, or, what is the same thing, the point of contact of this tangent. Suppose the tangent meets the conic U = (which is of course the conic corre- sponding to w = co)m the points P, P^, and let 0, oo be the parameters of the point P, and ff, 00 the parameters of the point P'. It follows from my "Note on the Geometrical representation of the Integral Jdx-i- V(aj + a) (a? + 6) (x + c)," [113] (*) and from the theory of invariants, that if nf represent the "Discriminant" of ^U + V ^ I take the opportonity of oorreoting an obvious error in the note in question, viz. a'+&'+c*-2&c-2eii-2a5 is throughout written instead of (what the expression should be) 6*c'+c'a*+a*&'-2a'&e-2&>ea-2c^. [This coiraotion is made, ante p. 55.] 88 NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCBIBED POLYGON. [ll5 (I now use the term discriminant in the same sense in which determinant is sometimes used, viz. the discriminant of a quadratic function cue* + iy* + c-g^' + %fyz + 2gzx + 2hxy or (a, 6, c, /, g, h) {x, y, zy, is the determinant k = abc — of* — 65^ — ch* + 2fgh), and if •^ 00 df then the following theorem is true, viz. "If {0y 00), (ffy 00) are the parameters of the points P, P' in which the conic U=0 is intersected by the tangent, the parameter of which is p, of the conic kU+V=0, then the equations n^ ^np-Uk, n^=np+nA:, determine the parameters 6, ff of the points in question." And again, — "If the variable parameters ^, ff are connected by the equation n^-n^=2nit, then the line Ff will be a tangent to the conic fcCr+F=0." Whence, also, — " If the sides of a triangle inscribed in the conic 17= touch the conies k I7+F=0, it'I7+F=0, rr7+F=o, then the equation nifc+nifc'+nr=o must hold good between the parameters fc, k\ VT And, conversely, when this equation holds good, there are an infinite number of triangles inscribed in the conic [7=0, and the sides of which touch the three conies ; and similarly for a polygon of any number of sidea The algebraical equivalent of the transcendental equation last written down is 1, k , Vn* 1, V, VqF = 0; 1, y\ VqF let it be required to find what this becomes when k^V ^kf' ^0^ we have 115] NOTE ON THE P0RI8M OF THE IN-AND-CIRCUMSCRIBED POLYGON. 89 and substituting these values, the determinant divides by 1, k\ ifc'» 1, r, r« the quotient being composed of the constant term C, and terms multiplied by k, k\ k" ; writing, therefore, fc = A/ = fc" = 0, we have (7=0 for the condition that there may be inscribed in the conic Z7 = an infinity of triangles circumscribed about the conic F=0; C is of course the coeflBcient of f" in Vnf> i-e. in the square root of the discriminant of f 17"+ F; and since precisely the same reasoning applies to a polygon of any number of sides, — Theorem. The condition that there may be inscribed in the conic CT^O an infinity of 7»-gons circumscribed about the conic F = 0, is that the coefficient of f*"' in the development in ascending powers of f of the square root of the discriminant of f[7+ F vanishes. [This and the theorem p. 90 are erroneous, see po»^, 116]. It is perhaps worth noticing that w = 2, i. e. the case where the polygon degene- rates into two coincident chords, is a case of exception. This is easily explained. In particular, the condition that there may be in the conic ^ flwr»+6y*-|-cz'=0 an infinity of »-gons circumscribed about the conic ic*+ y'+ -^' = 0, is that the coeflScient of 1^^ in the development in ascending powers of f of \/(l + a^)(l+6^)(l-hcf) vanishes ; or, developing each factor, the coefficient of f**~* in (l + iaf-Ja«f»-hT'ffa'?-?fia'^'+&c.)(l + i6f-&c.)(l + ic^-&c.) vanishea Thus, for a triangle this condition is a^ + 6» + c» - 26c - 2ca - 2a6 = ; for a quadrangle it is a» + 6* + c* - 6c«- 6»c - ca» - c^a - a6» - a'6 + 2a6c = 0, which may also be written (6 + c - a) (c + a — 6) (a + 6 - c) = ; and similarly for a pentagon, &c. ' I have, in order to present this result in the simplest form, purposely used a notation different from ^t of the note above referred to, the quantities ax>+&y*+c;e' and o^-h-y'^-^'Z^ being, in fact, interchanged. c. n. 12 90 NOTB ON THE PORISM OF THE IN-AND-OIRCUMSCRIBED POLYGON. [llo Suppose the conies reduce themselves to circles, or write i2 is of course the radius of the circumscribed circle, r the radius of the inscribed circle, and a the distance between the centres. Then f[7+F=(f+l, f+1, -fi?^i- + a«, 0, -a, 0)(x, y, 1)«, and the discriminant is therefore Hence, Theorem. The condition that there may be inscribed in the circle a^ + y* - ii*^ = an infinity of n-gons circumscribed about the circle (j?— a)*+ y* — r* = 0, is that the coefficient of f"""* in the development in ascending powers of f of may vanish. Now (A +Bf + 6'?)* = VJ |l + i5 1 + (i^C - i5«) f. + ...} . or the quantity to be considered is the coefficient of f^^ in (1 +*f-4p-) {i + ifi|+ (i^c- 45«)|;+ ...|,. where, of course, In particular, in the case of a triangle we have, equating to zero the coefficient of p, or substituting the values of A, B, C, (a» - i?)« - 4r»i? = 0, that is (a» -i?+ 2iJr)(a»-i?- 2iJ7) = ; the factor which corresponds to the proper geometrical solution of the question is a»-i? + 2iJr=0, Euler's well-known relation between the radii of the circles inscribed and circumscribed in and about a triangle, and the distance between the centres. I shall not now discuss the meaning of the other factor, or attempt to verify the formulae which have been given by Fuss, Steiner and Richelot, for the case of a polygon of 4, 5, 6, 7, 8, 9, 12, and 16 sides. See Steiner, CreUe, t. n. [1827] p. 289, Jacobi, t. in. [1828] p. 376 : Richelot, t. v. [1830] p. 250; and t. xxxviii. [1849] p. 353. 2 Stone Buildings, July 9, 1853. 116] 91 116. COREECTION OF TWO THEOREMS RELATING TO THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. [From the Philosophical Magazine, vol. VL (1853), pp. 376 — 377 ] The two theorems in my " Note on the Porism of the in-and-circumscribed Polygon " (see August Number), [115], are erroneous, the mistake arising from my having in- advertently assumed a wrong formulae for the addition of elliptic integrals. The first of the two theorems (which, in fact, includes the other as a particular case) should be as follows : — Theobem. The condition that there may be inscribed in the conic 17 = an infinity of n-gons circumscribed about the conic F=0, depends upon the development in ascending powers of f of the square root of the discriminant of fZ7+ F; viz. if this square root be then for n = 3, 5, 7, &c. respectively, the conditions are |C|=0, C, D D, E = 0, C, D, E D, E. F E, F, Q *nd for n = 4, 6, 8, &c respectively, the conditions are = 0, &C. ; ID 1=0, D, E E, F = 0, = 0, &c. D, E. F D, F, F, G, H ^6 examples require no correction; since for the triangle and the quadrilateral '^spectively, the conditions are (as in the erroneous theorem) (7 = 0, Z) = 0. 12—2 92 CORRECTION OF TWO THEOREMS RELATING TO THE PORISM, &C. [llG The second theorem gives the condition in the case where the conies are replaced by the circles ^r* + y* — i? = and (a? — a)* + y' — ^ = 0, the discriminant being in this case - (1 + f ) {r» + f (r« + i? - aO + f^iP}. As a very simple example, suppose that the circles are concentric, or assume a = ; the square root of the discriminant is here (l+f)\/r» + i?f; and putting for shortness -^ = «» we may write il+5f+...=(l+f)VrTif, that is, ^=1, fi = ia + l. C = -ia» + ia», 2) = tV«'-*«"> ^ = -TfF«* + TV«'> &c. ; thus in the case of the pentagon, C^-D» = j^a*{(a-4)(5a-8)-4(a-2)«l = tAi «*(«*- 12a + 16), and the required condition therefore is a»-12a + 16 = 0. It is clear that, in the case in question, -^ = cos36° = i(^5 + l), that is, - = V5-1, or (12 + r)«-5r«= 0, viz. (Va + 1)" - 5 = 0, or a + 2 Va- 4 = 0, the rational form of which is (^- 12a + 16 = 0, and we have thus a verification of the theorem for this particular case. 2 Stone Buildings, Oct. 10, 1853. 117] 93 117. NOTE ON THE INTEGRAL Idx^Jim-^x) (x + a){x + b){x + c). [From the Philosophical Magazine, vol. vi. (1853), pp. 103 — 105.] If in the formubB of my " Note on the Porism of the in-and-circumscribed Polygon/' [115], it is assumed that and if a new parameter cd connected with the paramieter w by the equation com w= m — G) be made use of instead of w, then and thus the equation wU+V=Oy viz. the equation G)(d5» + y" + -2^) + aa:' + 6y« + C2» = 0, is precisely of the same form as that considered in my "Note on the Geometrical Representation of the Integral Idx -=- V(a: + «)(« + 6) {x + c)," [113.] Moreover, introducing instead of f a quantity 17, such that then VDf "Jim-ri) (a + 17) (6 + 17) (c + 17)' 94 NOTE ON THE INTEGRAL j dx-i- J{m-'X) (x + o) (x + b) (x + c). [ll7 Also f = 00 gives 17 = m, the integral to be considered is therefore 11,17=1 . -- — ; Jm v(m-i7)(a + i7)(D + i7)(c + i7) i.e. if in the paper last referred to the parameter 00 had been throughout replaced by the parameter m, the integral drj ni7=f -j= V(a 4- 17) (6 + 17) (c + 17) would have had to be replaced by the integral 11,17. It is, I think, worth while to reproduce for this more general case a portion of the investigations of the paper in question, for the sake of exhibiting the rational and integral form of the algebraical equation corresponding to the transcendental equation ±Tl,k±TI,p ±11,0=^0, Consider the point f, 17, f on the conic m(j:' + y' + ^) + aaj"-f fty" + C£i" = 0, the equation of the tangent at thife point is (m + a) f a? + (m + 6) i7y + (m + c) 5? = ; and if ^ be the other parameter of this line, then the line touches ^ (a;» -f y " + ^) + OA-" + 6y> + c^ = ; or we have d + a "^ ^ + 6 ^ + c ~ ' and combining this with (m + a)f" + (m + 6)i7« + (m + c)f = 0, we have f : 17 : g'= ^ b "C 'Ja-^-d "Jb -k-m ^c+m : ^{c-'a)'Jb + 0^c -{•m'Ja + m : V(a '-b)'^C'\-0^a + m^b'k-m for the coordinates of the point P. Substituting these for x, y, z in the equation of the line PP" (the parameters of which are p, k\ viz. in a?V6--c v^(a4- A;)(a+p) + yVc-aV(6 + A;)(6+p) + 2:Va-6Vc + fcVc+;)=0, we have 'Ja + m V6 + m vc + m 117] NOTE ON THE INTEGRAL I dx-^ J{m-'X) (x + o) (x + b) (x + c). 95 which is to be replaced by (a+p)(a-h^)(a + g)^ 6 + m These equations give, omitting the common factor (a + wi) (6 + m) (c + m), + {abc {kd-\-0p+ kp) -\-pk6 {bc-{-ca + a6)j, -{■m[''aJbc -pkO-^ (bc + ca-k-ab) (/> + *+ 5) + (*^ + ^[p +;>*)(« + 6 + c)} + {oftc (p + ^* + ^) - pk0 (a + 6 + c)} , + m {{be -{-ca-^- ab) - {k6 + 6!p +#)} + ahc+pkO; and substituting in 4X*. ft' — (2X/a)* = 0, we have the relation required. To verify that the equation so obtained is in fact the algebraical equivalent of the transcendental equation, it is only necessary to remark, that the values of \\ /i' are unaltered, and that of \fi only changes its sign when a, 6, c, m and />, k, 0^ —m are interchanged ; and so this change will not affect the equation obtained by substituting in the equation 4X' . ^' — (2X/a)' = 0. Hence precisely the same equation would be obtained by eliminating L, M from (ifc + a) (ifc + 6) (ifc + c) = (Z + Jfit)» (m - A), (p + a)(p + 6)Cp + c) = (Z + ifi>)*(m-p), (^ + a)(5 + 6)(d+c) = (Z + Jftf)«(d-j9); or, putting (Z + Mk) (m— k)='a + ^k + ^h?, by eliminating a, ^, 7 from (m - k){k + a) (k + 6) (i + c) = (a + /3A + yk'Y, (m-p)(p + a)(/> + 6)(/> + c) = (a + /3p + 7p«)«, (m-5)(5+a)(5+6)((9+c) = (a + /9^ + 7^)^ = (a + ^m + 7m')«, which by Abel's theorem show that p, i, ^ are connected by the transcendental equation above mentioned. 2 Stone BuildingSy July 9, 1853. 96 [118 118. ON THE , HAEMONIC RELATION OF TWO LINES OR TWO POINTS. [From the Philosophical Magaziney vol. VI. (1863), pp. 103 — 107.] The "harmonic relation of a point and line with respect to a triangle'* is well known and understood ^ ; but the analogous relation between two lines with respect to a quadrilateral, or between two points with respect to a quadrangle, is not, I think, sufficiently singled out from the mass of geometrical theorems so as to be recognized when implicitly occurring in the course of an investigation. The relation in question, or some particular case of it, is of frequent occurrence in the Traits des ProprOtis Projectives, [Paris, 1822], and is, in fact, there substantially demonstrated (see No. 163); and an explicit statement of the theorem is given by M. Steiner, Lehrsatze 24 and 25, Crelle, t. xiil [1835] p, 212 (a demonstration is given, t. xix. [1839] p. 227). The theorem containing the relation in question may be thus stated. Theorem of the harmonic relation of two lines with respect to a qvadrUaieral. " If on each of the three diagonals of a quadriUiteral there be taken two points harmonically related with respect to the angles upon this diagonal, then if three of the points lie in a liney the other three points will also lie in a line^* — the two lines are said to be harmonically related with respect to the quadrilateral. It may be as well to exhibit this relation in a somewhat different form. The three diagonals of the quadrilateral form a triangle, the sides of which contain the six angles of the quadrilateral; and considering three only of these six angles (one angle on each side), these three angles are points which either lie in a line, or else ^ The relation to which I refer is contained in the theorem, *'If on each side of a triangU there be taken two points harmonicaUy related with respect to the angles on this side, then if three of these points lie in a line, the lines joining the other three points with the opposite angles of the triangle meet in a point,** — the line and point are said to be harmonically related with respect to the triangle. 118] ON THE HARMONIC RELATION OF TWO LINES OR TWO POINTS. 97 are such that the lines joining them with the opposite angles of the triangle meet in a point. Each of these points is, with respect to the involution formed by the two angles of the triangle, and the two points harmonically related thereto, a double point; and we have thus the following theorem of the harmonic relation of two lines to a triangle and line, or else to a triangle and point. Theorem. '' If on the sides of a triangle there be taken three points, which either lie in a line, or else are such that the lines joining them with the opposite angles of a triangle meet in a point; and if on each side of the triangle there be taken two points, forming with the two angles on the same side an involution having the first-mentioned point on the same side for a double point ; then if three of the six points lie in a line, the other three of the six points will also lie in a line," — the two lines are said to be harmonically related to the triangle and line, or (as the case may be) to the triangle and point. The theorems with respect to the harmonic relation of two points' are of course the reciprocals of those with respect to the harmonic relation of two lines, and do not need to be separately stated. The preceding theorems are useful in (among other geometrical investigations) the porism of the in-and-drcumscribed polygon. 2 Stone Buildings, July 9, 1863. C. n. 13 98 [119 119. ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. [From the Philosophical Magnziney vol. vi. (1853), pp. 182—185.] The theorem to which I refer is remarkable for the extreme simplicity of its demonstration. Let it be required to expand the factorial x—a x — h x-^c ,.. in the form x — a x — x — y...-^- Bx" a a?--^... + Oa? — a ... + D ... &c. We have first a: — a = ar — a + a — a; multiply the two sides of this by a: — 6 ; but in multiplying by this factor the term .'/; — a, write the factor in the form a? — ^ + iS — 6 ; and in multiplying the term a — a, write the factor in the form x — a + a — b; the result is obviously x — ax— h^x — ax — fi +(a — a + /8 — 6)a? — a +a— aa— 6; multiply this by a; — c, this factor being in multiplying the quantity on the right-hand side written successively under the forms a: — 7 + 7 — c, a?— /8-I-/8 — c, a — a-^a — c] the result is X — a x^b X — c= x — ax — fix — y + (a — a-hyS — 6 + 7 — c)a: — aa: — /8 + (a — a a— 6 + a-a /S — c-^-fi — b jS — c) x - a 119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 99 which may be thus written, (x — a) (a? — 6) (ar — c) = Consider, for instance, La, 6, cJ, then, paying attention in the first instance to the Greek letters only, it is clear that the terms on the second side contain the combinations two and two, with repetitions, of the Greek letters a, ^, and these letters appear in each tenn in the alphabetical order. Each such combination may therefore be considered as derived from the primitive combination a, a by a change of one or both of the as into ^; and if we take (instead of the mere combination a, a) the complete first term a — a a — 6, and simultaneously with the change of the a of either of the factors into ^ make a similar change in the Latin letter of the factor, we derive from the first term the other terms of the expression on the right-hand side of the expression. It is proper also to remark, that, pa3dng attention to the Latin letters only, the different terms contain all the combinations two and two, without repetitions, of the letters a, 6, c. The same reasoning will show that x — ax — bx — cx — d== x-^a x — x — y x — 8 La, 6, c, aJj la, b, c, dJ, [a. /8 -| + x — a La, 6, c, dJ, La, b , c, dJt where, for instance, La, b, c, dit +(a-a)(a-b)(^-d) + (a-a)(/8-c)(/9-d) + (/9-6)(/3-c)(/9-d),&e. 13—2 100 ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. [119 It is of course easy, by the use of subscript letters and signs of summation, to present the preceding theorem under a more condensed form ; thus writing LC&i U] ... Of , . . (lf.^gj f ^1 ^ ) where k,, Av-i, ...&o form a decreasing series (equality of successive terms not excluded) of numbers out of the system r, r — 1, ...3, 2, 1; the theorem may be written in the form X but I think that a more definite idea of the theorem is obtained through the notation first made use o£ It is clear that the above theorem includes the binomial theorem for positive integers, the corresponding theorem for an ordinary factorial, and a variety of other theorems relating to combinationa Thus, for instance, if Gg(ai,...ap) denote the combinations of a,, ... Op, q and q together without repetitions, and ^g (Oi, ... Op) denote the combinations of a,, ... Op, q and q together with repetitions, then making all the a's vanish, and therefore a?- tti ... aj-Op = ^^o(-)' C'gCoi* ••• ap)«*^r (X - a)P = SfoH' C, (a, a . . . plures) a^ = ^ S H' H a^ ^^. the ordinary binomial theorem for a positive and integral index p. So making all the a's vanish, a?** = /S^^o-ff9(«i ••• «p-«+i) ^^ *i ^"" ^ •••*■" "p-«- If m be any integer less than p^ the coefl&cient of x^ on the right-hand side must vanish, that is, we must have identically So also ffti . . . ci^«^+i I Cp-^((h, a,, ... Op) = Sqo (-)«Ci_^M»(«i» «» ••• «P-«) [^ ^J^- Suppose 01 = 0, a, = l ...ap=2) — 1; ax=A, o, = i:— 1,... ap«fc-2)+ 1, then [:':..^J.-[":!.:.!:;:a-t^t*''^-'"^^-'''"^ and hence the binomial theorem for factorials. 119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 101 A preceding formula gives at once the theorem It may be as well to remark^ with reference to a demonstration frequently given of the binomial theorem, that in whatever way the binomial theorem is demonstrated for integer positive indices, it follows from what has preceded that it is quite as easy to demonstrate the corresponding theorem for the factorial [m]^. But the theorem being true for the factorial [m]^, it is at once seen that the product of the series for (l+xy* and (1 +«?)** is identical with the series for (l+a:)*^+^ and thus it becomes unnecessary to employ for the purpose of proving this identity the so-called principle of the permanence of equivalent forms ; a principle which however, in the case in question, may legitimately be employed. 102 [120 120. NOTE ON A GENERALIZATION OF A BINOMIAL THEOREM. [From the Philosophical Magazine, vol. vi. (1853), p. 185.] The formula {CrelUy t. i. [1826] p. 367) for the development of the binomial {x + a)*', but which is there presented in a form which does not put in evidence the law of the coefficients, is substantially equivalent to the theorem given by me as one of the Senate House Problems in the year 1851, and which is as follows : — "If {a 4-^8 + 7...)'' denote the expansion of (a 4-^ + 7... )p, retaining those terms Na^ff^rfh^ ,,. only in which 6 + c + d... is not greater than /) — 1, c-frf-f.. is not greater than p — 2, &c., then ar** = 1 (a: + o)** - "^"7'2W^^ {«-h/3-h7N^-i-a4-/3-h7 + 8)-. + &c.' >i The theorem is, I think, one of some interest. 121] 103 121. NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. [From the Philosophical Magazine^ vol. vi. (1853), p. 259.] The following question was suggested to me, either by some of Prof Boole's memoirs on the subject of probabilities, or in conversation with him, I forget which ; it seems to me a good instance of the class of questions to which it belongs. Given the probability a that a cause A will act, and the probability p that A acting the effect will happen; also the probability /8 that a cause B will su^t, and the probability q that B acting the effect will happen; required the total probability of the effect. As an instance of the precise case contemplated, take the following: say a day is called windy if there is at least w of wind, and a day is called rainy if there is at least r of rain, and a day is called stormy if there is at least W of wind, or if there is at least R of rain. The day may therefore be stormy because of there being at least W of wind, or because of there being at least R of rain, or on both 8w;counts ; but if there is less than W of wind and less than R of rain, the day will not be stormy. Then a is the probability that a day chosen at random will be windy, p the probability that a windy day chosen at random will be stormy, /8 the probability that a day chosen at random will be rainy, q the probability that a rainy day chosen at random will be stormy. The quantities X, fi introduced in the solution of the question mean in this psurticular instance, \ the probability that a windy day chosen at random will be stormy by reason of the quantity of wind, or in other words, that there will be at least W of wind; fi the probability that a rainy day chosen at random will be stormy by reason of the quantity of rain, or in other words, that there will be at least R of rain. 104 NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. [l21 The sense of the terms being clearly understood, the problem presents of course no difficulty. Let X be the probability that the cause A acting will act efficaciously ; fi the probability that the cause B acting will act efficaciously; then P = \ + (1-X)aa^. g = /[A + (l-/Lt)aX, which determine \, fi] and the total probability p of the effect is given by p = Xa + ^ - X/itt/S ; suppose, for instance, a=l, then ;? = X + (l-X)/i^, g = /Lt+ X-X/Li, p = X + /Lt^-Xft/8, that is, p = p, for p is in this case the probability that (acting a cause which is certain to act) the effect will happen, or what is the same thing, p is the probability that the effect will happen. Machynlleth, August 16, 1853. 122] 105 122. ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE OF THE SECOND ORDER INTO ITSELF. [From the Philosophical Magazine, vol. vi. (1853), pp. 326—333.] The following theorems in plane geometry, relating to polygons of any number (odd or even) of sides, are well known. " If there be a polygon of (m + 1) sides inscribed in a conic, and m of the sides pass through given points, the (m + l)th side will envelope a conic having double coutact with the given conic." And " If there be a polygon of (m + 1) sides inscribed ^ a conic, and m of the sides touch conies having double contact with the given ^nic, the {m + l)th side will envelope a conic having double contact with the given conic.*' The second theorem of course includes the first, but I state the two separately for the sake of comparison vdth what follows. As regards the corresponding theory in geometry of three dimensions, Sir W. Hamilton given a theorem relating to polygons of an odd number of sides, which may be thus stated: "If there be a polygon of (2m + 1) sides inscribed in a surface of the ^^ud order, and 2m of the sides pass through given points, the (2m + l)th side will constantly touch two sur&ces of the second order, each of them intersecting the given 8urfece of the second order in the same four lines ^" ^ See Phil, Mag. vol xxzt. [1S49] p. 200. The form in which the theorem is exhibited bj Sir W. Hamilton u somewhat different ; the surface containing the angles is considered as being an ellipsoid, and the two sorfaoes ^'^'i^ hj the last or (2]ii + l)th side of the polygon are spoken of as being an ellipsoid, and a hjperboloid of two sheets, having respeotivelj doable contact with the given ellipsoid : the contact is, in fact, a qoadraple con- ^ ^ the same four points ; real as regards two of them in the case of the ellipsoid, and as regards the other ^ ift the case of the hyperboloid of two sheets ; and a qoadraple contact is the coincidence of foar generating ^^>Mi belonging two and two to the two series of generating lines, these generating lines being of coarse (in the ^Me Qoniidered by Sir W. Hamilton) all of them imaginary. c. n. 14 106 ON THE HOMOQRAPHIC TBANSFOBMATION OF [122 The entire theory depends upon Tvhat may be termed the transformation of a sur£BM^ of the second order into itself, or analytically, upon the transformation of a quadratic form of four indeterminates into itself I use for shortness the term trans^- formation simply; but this is to be understood as meaning a homographic transformation, or in analytic language, a transformation by means of linear substitutions. It will be convenient to remark at the outset, that if two points of a surface of the second order have the relation contemplated in the data of Sir W. Hamilton's theorem (viz. if the line joining the two points pass through a fixed point), the transformation is, using the language of the Recherchss Arithm^tiqtiea, an improper one, but that the relation cont-emplated in the conclusion of the theorem (viz. that of two points of a surface of the second order, connected by a line touching two surfaces of the second order each of them intersecting the given sur&ce of the second order in the same four lines) depends upon a proper transformation; and that the circumstance that an even number of improper transformations is required in order to make a proper trans- formation (that this circumstance, I say), is the reason why the theorem applies to polygons in which an even number of sides pass through fixed points, that is, to polygons of an odd number of sides. Consider, in the first place, two points of a sur£ace of the second order such that the line joining them passes through a given point. Let a, y, z, w he current coordinates S and let the equation of the surface be (a,. ..)(«, y, z, wy = 0, and take for the coordinates of the two points on the surface Xi, yi, Zi, v\ and ^i> Vti ^s> ^1} ai^d for the coordinates of the fixed point a, /3, 7, S. Write for shortness (a, ...)(«, /S, 7, S)«=p, (a, ...)(«, ^, 7, S)(a?i, yi, z^, Wi) = qu then the coordinates ^, y,, z^, w^ are determined by the very simple formulsB 2ce 27 P 2S ^ Strictly speaking, it is the ratios of these qoantities, e.g. « : tr, y : 19, z : w, whioh are the ooordinates, and consequently, even when the point is given, the values «, y, «, to are essentially indeterminate to a factor prh. So that in assu mi ng that a point is given, we should write xiyiz: w=a : P'.yiH; and that when a point is obtained as the result of an analytical process, the conclusion is necessarily of the form just mentioned : but when this is once understood, the language of the text may be properly employed. It may be proper to explain here a notation made use of in the text: taking for greater simplicity the case of forms of two variables, (I, m) {x^ y) means te+my ; (a, 6, e) («, y)« means a«*+26xy+cy«; {a,b,e) ft, iy) (ar, y) means a^x + h{fy + rtx) + e7fy. The system of coefficients may frequently be indicated by a single coefficient only : thus in the text (a, ...) («, y, f , ir)^ stands for the most general quadratic ftmotion of four variables. 122] A SUBFACE OF THE SEOOND ORDER INTO ITSELF. 107 In &ct, these values satisfy identically the equations «i, y%, ^, w% 0, «i, Vu Sk, Wi a, 0, y, S that is, the point (xt, y%, z^, w^ will be a point in the line joining {Xi, j/i, z^ Wi) and (a, fi, y, S). Moreover, (a, ...)(«!, yi» -^1, w«)*= (a» ...)(«^i. Vu ^» ^y --^ (a, ...)(«, A 7, S)(«i, yi, Zu Wi) (a, ...) («i, yi, ir^, t^i)« - -^' ?i + % i>, that is, 80 that d?|, yi, Zi, Wi being a point on the surface, a?„ y,, z^, w^ will be so too. The equation just found may be considered as expressing that the linear equations are a transformation of the quadratic form (a, ...)(a;, y, z, vif into itself If in the system of linear equations the coefficients on the right-hand side were arranged square-wise, and the determinant formed by these quantities calculated, it would be found that the value of this determinant is —1. The transformation is on this account said to be improper. If in a system of linear equations for the transformation of the form mto itself the determinant (which is necessarily -h 1 or else — 1) be +1, the trans- formation is in this case said to be proper. We have next to investigate the theory of the proper transformations of a quadratic form of four indeterminates into itself This might be done for the absolutely general form by means of the theory recently established by M. Hermite, but it will be sufficient for the present purpose to consider the system of equations for the trans- f(Mination of the form aj*-hy' + -2^ + ti;* into itself given by me some years since. (Crdle, voL xxxn. [1846] p. 119, [52] Q). I proceed to establish (by M. Hermite's method) the formulae for the particular case in question. The thing required is to find ^, y„ z^, w^ linear functions of "hf Vu ^t ^u such that ^' + y«* + z^^ -I- 1^,« = a?i* + yi« + -^1* + Wi*. Write flr,+«i=2f, yi + y, = 2i7, Zi + z^:=2Z ii;i + w, = 2o); ^ It is a jjingnlar instanoe of the way in which different theories connect themselves together, that the toannls in qoeetion were gooeralizations of Baler's formolaB for the rotation of a solid body, and also are loaaavlm which reappear in the theory of qaatemions ; the general formolas cannot be established by any obyions ggpsraliaation of the theory of qaatemions. 14—2 108 ON THE HOMOGRAPHIC TRAJ^SFOEMATION OF [122 then putting a:, = 2f — a?i, &c., the proposed equation will be satisfied if only f" + if + (7+ «*= f^ + Wi + f^i + a)Wi, which will obviously be the case if Zi = /Ltf — Xiy + ? + ccD , Wi = — af — 617 — c(^ + a , where X, /li^ 1^, a, b, c are arbitrary. Write for shortness a\ + bfjL + cv^il>, l + X« + /Lt« + i/» + a« + 6' + (^ + 0« = A;, then we have A:f = (l +X*+ 6"+ (^)a^ + (XM-y -a5-c0)yi + (i'X. + /i-ca + 6^)^i + (6i;-c/i-a-\^)wi, A:(;' = (i'X-/Lt-ca-6^)a?i + (AM/ + X — 6c +a^)yi+(l +i^ + a*+fe» )-?!+ (aft—ftX.- 0-1/^)1^1, A;u;= (61/ - c/i+ a +\^) a?i + (cX. -ay+ 6 +/A^)yi+ (a^ -6*'+ c + 1/^) ^1 + (1 + X* +/Lt» + i^) i^/i ; and fix>m these we obtain Auji = (1 + X» + 6« + (^ - /Lt*- 1/'- a' - ^)a?i + 2 (X/Li-i/- oft- c<^)yi + 2 (vX. + M-ca + 6^) ^1 + 2 (6v — c/Li — a— X^) Wi, A:ya = 2(X/Lt + y-a6 + c0)a?i + (l + M* + (^ + a«-i^-X«-fe«-<^)yi4-2(/[Ay-X-6c-a^)^^ + 2 (cX — ai/ — 6 — /Lt^) w/j , A:-?a = 2(i/X-/Lt-ca-6^)a?i + 2(/Ay + X-6c + a^)yi + (l + i/» + a» + 6*-X«-/Lt*-c>-«^)^^ + 2 (a/Li — 6X — c — y^) Wi , Arwj = 2 (61/ — c/Li + a + X^) a?i + 2 (cX - ai/ + 6 + ft^) yi + 2 (a/Li — 61/ + c + p4^)^i + (1 + X» +/iA« + i;»-a« -6» - c"- 0«) Wi, values which satisfy identically x^ -V yf •\' zf •\- w^ ^ x^ •\- y^ '\- z^ •\- w^. Dividing the linear equations by &, and forming with the coefficients on the right- hand side of the equation so obtained a determinant, the value of this determinant is 4- 1 ; the transformation is consequently a proper one. And conversely, what is very important, every proper transformation may be exhibited under the preceding form*. I The nature of the reasoning by which this is to be established may be seen by considering the analogous relation for two variables. Suppose that x^, y^ are linear functions of x and y such that x^-{-y^=a^+y'*\ then if 2f=« + ari, 2iy=y+yi, ^, 17 will be linear functions of ar, y such that ?+i;*=$«+i;y, or $(f-«) + i7(i7-y) = 0; ^-x must be divisible either by ri or else by 17-y. On the former supposition, calling the quotient y, we have :c=(-Fi}, and thence ^=^(+17, leading to a transformation such as is considered in the text, and which is a proper transformation; the latter supposition leads to an improper transformation. The given transformation, assumed to be proper, exists and -cannot be obtained from the second supposition; it must therefore be obtainable from the first supposition, i.e. it is a transformation which may be exhibited under a form such as is considered in the text. 122] A SURFACE OF THE SECOND ORDER INTO ITSBLF. 109 Next considering the equations connecting x, y, z, w with {, 97, (^, od, we see that + ( Mf-A'^+ ?+Cft))» + (-af -617- cf+ ft))*. We are thus led to the discussion (in connexion with the question of the trans- formation into itself of the form a" + y* + ^ + ti;*) of the new form + (--vx+ y + 7iz + bvif + ( /jLx — \y+ z + cwy + (— oo? — 6y — c-^4- w)"; or, as it may also be written, (af + y* + j^-{'tul') + (vy-fiz-{'awy-{-(\z — vx + bwy + (jix-\y-\'C^ Represent for a moment the forms in question by U, F, and consider the surfaces [7=0, F=0. If we form from this the sur&ce V+qU=0, and consider the dis- cnminant of the function on the left-hand side, then putting for shortness /c = V + /Lt« + i/« + a« + 6»+c>, this discriminant is which shows that the sur&ces intersect in four lines. Suppose the discriminant vanishes; we have for the determination of 9 a quadratic equation, which may be written 5« + (2 + /e)5 + ir=0; let the roots of this equation be q,, q,/, then each of the functions q,U-\-V, q,,U-^ V will break up into linear factors, and we may write q,U+V^RA> q,,U+r^RAr (U and V are of course linear functions of R^, and R„8,^) forms which put in evidence the fact of the two sur&ces intersecting in four lines. The equations a^ + a?, = 2f, yi+yt«2i;, Zi+Zi^2^, Wi + t£;, = 2fii, no ON THE HOMOGRAPHIC TRANSFORMATION OF [122 show that the point (f, 17, f, «) lies in the line joining the points (a^i, yi, z^ Wi) and (^11 3/21 ^21 wjj); aiid to show that this line touches the surface F=0, it is only necessary to form the equation of the tangent plane at the point (f, 1;, (Ti <») o{ the 8ur£BU3e in question ; this is (x + vy --fjLZ + at(;) (f + V17 — ft(r + ck») + ... = ; or what is the same thing, {x •\- vy "' fiz •¥ aw) a?i + . . . = 0, which is satisfied by writing (a?,, yi, i^i, Wj) for (x, y, -e, w), that is, the tangent plane of the surface contains the point (xi, y^ Zi, w,). We see, therefore, that the line through (^, yii ^1. «^i) aiid (^> yjj -8^2 » ^2) touches the surface F=0 at the point (f, 17, f, a>). Write now ^=~a'' ^=~r"' ^=":r» ^=^"a"» f^'^'iT' ^="^5 <p 9 9 9 9 9 if we derive from the coordinates a?!, yi, ^Ti, Wi, by means of these coefficients a\ b\ c\ \'y f/, v\ new coordinates in the same way as «i, y„ z^ w^ were derived by means of the coefficients a, 6, c, X, /li, r, the coordinates so obtained are — a?i, — yi,~-z^i, — Wi, i.e. we obtain the very same point (x^, y,, ^si ^i) by means of the coefficients (a, 6, c, X, /a, i;), and by means of the coefficients (a\ h\ c\ \\ /li', v^). Call f, rf, f, oi' what f, 17, f, c» become when the second system of coefficients is substituted for the first; the point ^j V> (r'» ^' will be a point on the sur&oe F' = 0, where F'=<^>(a;» + y« + ^ + ti;») + (— cy + 62: — Xw)* + (— 0^ -i: ca? — /it^;)* + (— 6a: + ay — j/u;)" + (— Xr — /Lty — i/^)* ; and since F+ F' = /e(a» + y« + j^» + t£;»), and F=0 intersects the surface iB* + y* + ^ + tt;* = in four lines, the surface F' = will also intersect this surface in the same four lines. And it is, moreover, clear that the line joining the points {x^, yi, Zi, Wi) and (x^, y,, ^„ n;,) touches the surfieu^ F' = in the point (f', 17', f, co'). We thus arrive at the theorem, that when two points of a surfisu^ of the second order are so connected that the coordinates of the one point are linear functions of the coordinates of the other point, and the transformation is a proper one, the line joining the two points touches two sur&ces of the second order, each of them intersecting the given surface of the second order in the same four lines. Any two points so connected may be said to be corresponding points, or simply a pair. Suppose the four lines and also a single pair is given, it is not for the determination of the other pairs necessary to resort to the two auxiliary surfaces of the second order ; it is only necessary to consider each point of the surface as determined by the two generating lines which pass through it; then considering first 122] A SURFACE OF THE SECOND ORDER INTO ITSELF. Ill one point of the given pair, and the point the corresponding point to which has to be determined, take through each of these points a generating line, and take also two generating lines out of the given system of four lines, the four generating lines in question being all of them of the same set, these four generating lines inter- secting either of the other two generating lines of the given sjrstem of four lines in four pointa Imagine the same thing done with the other point of the given pair and the required point, we should have another system of four points (two of them of course identical with two of the points of the first-mentioned system of four points); these two systems must have their anharmonic ratios the same, a condition which enables the determination of the generating line in question through the required point: the other generating line through the required point is of course determined in the same manner, and thus the required point (i.e. the point corresponding to any point of the surface taken at pleasure) is determined by means of the two generating lines through such required point. It is of course to be understood that the points of each pair belong to two distinct systems, and that the point belonging to the one system is not to be con- founded or interchanged with the point belonging to the other system. Consider, now, a point of the surface, and the line joining such point with its corresponding point, but let the corresponding point itself be altogether dropped out of view. There are two directions in which we may pass along the surface to a consecutive point, in such mamier that the line belonging to the point in question may be intersected by the line belonging to the consecutive point. We have thus upon the surface two series of curves, such that a curve of each series passes through a point chosen at pleasure on the surface. The lines belonging to the curves of the one series generate a series of developables, the edges of regression of which lie on one of the surfaces intersecting the surfeice of the second order in the four given lines; the lines belonging to the curves of the other series generate a series of developables, the edges of regression of which lie on the other of the surfaces intersecting the surface in the four given lines; the general nature of the system may be understood by considering the system of normals of a surface of the second order. Consider, now, the surface of the second order as given, and also the two surfaces of the second order inter- secting it in the same four lines ; from any point of the sur&ce we may draw to the auxiliary surfiEtces four dififerent tangents ; but selecting any one of these, and considering the other point in which it intersects the surface as the point corre- sponding to the first*mentioned point, we may, as above, construct the entire system of corresponding points, and then the line joining any two corresponding points will be a tangent to the two auxiliary surfSaces; the system of tangents so obtained may be called a system of congruent tangents. Now if we take upon the surface three points such that the first and second are corresponding points, and that the second and third are corresponding points, then it is obvious that the third and first are corresponding points ;— observe that the two auxiliary surfaces for expressing the corre- spondence between the first and second point, those for the second and third point, and those for the third and first point, meet the surface, the two auxiliary surfaces of each pair in the same four lines, but that these systems of four lines are different 112 ON THB HOMOGRAPHIC TRANSFOBMATION OF A SURFACE &C. [122 for the different pairs of auxiliary surfaces. The same thing of course applies to any numb^ of corresponding points. We have thus, finally, the theorem, if there be a polygon of (m + 1) sides inscribed in a surfiEtce of the second order, and the first side of the polygon constantly touches two surfaces of the second order, e8w;h of them intersecting the surfisu^ of the second order in the same four lines (and the side belong always to the same system of congruent tangents), and if the same property exists with respect to the second, third, &c.... and wth side of the polygon, then will the same property exist with respect to the (m + l)th side of the polygon. We may add, that, instead of satisfying the conditions of the theorem, any two consecutive sides of the polygon, or the sides forming any number of pairs of con- secutive sides, may pass each through a fixed point. This is of course only a particular case of the improper transformation of a surfietce of a second order into itself, a question which is not discussed in the present paper. 1231 113 123. ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [From the Philosophical Magazine, vol. vi. (1853), pp. 414 — 418.] The equation of a surface passing through the curve of intersection of the surfaces a^+ y*+ -^'+ u;* = 0, flw^ + 6y* + cz^ H- dv)* = 0, is of the form »(a^ + y' + -2' + t£;') + aa^ + 6y' + c^ + dti;* = 0, where tt is an arbitrary parameter. Suppose that the sur&ce touches a given plane, we have for the determination of 8 a cubic equation the roots of which may be considered as parameters defining the plane in question. Let one of the values of 8 be considered equal to a given quantity k, the plane touches the surface aud the other two values of 8 may be considered as parameters defining the particular tangent plane, or what is the same thing, determining its point of contact with the Burface. Or more clearly, thus: — in order to determine the position of a point on the 8ur£BM^e A; (a;* + y2 + ^* + w") + aa?» + 6y« + C2:« + dt(;» = ; the tangent plane at the point in question is touched by two other surfaces /> (^ + y ' + ^ + 1^;") + cue" + 6y * + c^* + dw" = 0, 9(«' + y* + '8^* + w*) + cuc» + 6y' + C2:* + dt£;» = 0; c. n. 15 114 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [l23 and, this being so, p and q are the parameters by which the point in question is determined. We may for shortness speak of the surface A: (iB» + y» + ^ + w") + oa^ + 6^ + c-e' + dw" = as the surfiiK^ (jk). It is clear that we shall then have to speak of ai^ + y^ + z^ + v)^ = as the surface (oo). I consider now a chord of the surface (oo) touching the two surfaces (k) and (k')] and I take 0, <l> aa the parameters of the one extremity of this chord; (p, q) as the parameters of the point of contact with the surface (A;); p', q' as the parameters of the point of contact with the surface (kf); and O', ^' as the parameters of the other extremity of the chord; the points in question may therefore be distinguished as the points (oo ; ^, 0), (i; p, g), (k' \ p\ g'), and (oo; ^, ^'). The coordinates of the point (oo ; 0, <f>) are given by X : y : z : w= V(a + ^) (a + 0) -:- V(a — 6) (a — c) (a - d) V(6+^)(6 + </>) -r V(6-c)(6-d)(6-a) V(c + ^) (c + 4>) -r V(c - d) (c - a) (c - 6) ^/(dTW(d + 'f) H- V(d-a)(d-6)(d-c) ; those of the point (k; p, q) by X ', y : z : w^ V(a +p) (a + j) -^ V(a — 6) (a — c) (a — d) Va + i V(6+p)(6 + 5) -^ V'(6-c)(6-d)(6-a) ViTib V(c + p) (c + g) -r V(c - d) (c - a) (c - 6) Vc+i V(d+/>)(d + g)-f.V(d-a)(d-6)(d-c) Vd+l; and similarly for the other two points. Consider, in the first place, the chord in question as a tangent to the two surfaces (£) and {kf). It is clear that the tangent plane to the surface {k) at the point {k\ p, q) must contain the point (Ar'; p\ <f)^ and vice versd. Take for a moment f, i;, (r> ® ®* *^® coordinates of the point (A; p, g), the equation of the tangent plane to (£) at this point is 2(a + A:)fd: = 0; or substituting for f,... their values S (a;V(a+p)(a + g) Va -{Ic -r- V(a - 6) (a - c) (a - d) ) = ; 123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELTAN INTEGRAL. 115 or taking for a?,... the coordinates of the point (kf^ p\ ^\ we have for the conditions that this point may lie in the taDgent plane in question, or under a somewhat more convenient form we have 2((6-c)(c-d)(d-6)V(a+p)(a + ?)V(a + p')(a4-gO^^) = 0, for the condition in order that the point (Je\ p\ q') may lie in the tangent plane at (^> P* ?) ^ *^® surface {k). Similarly, we have 2f(6-c)(c-cO(d-6)V(a+i>)(a + g)V(a+/)0(a + 9')^5^')=O, \ ya-\- kJ for the condition in order that the point (A?, p, q) may lie in the tangent plane at ifff\ p\ 5O ^ *^® surfiwje Qf). The former of these two equations is equivalent to the sjrstem of equations V(a +/>) (a + q) (a +;>0 (« + 9') V l^A/ = ^ + A^ + ^'> and the latter to the system of equations V(o +p) (o + q) (a + p') (a + q) y |^ = V + /t'o + v'a*; where in each system a is to be successively replaced by 6, c, d, and where X, /ia, 1/ and V, ;i', 1/ are indeterminate. Now dividing each equation of the one system by the corresponding equation in the other system, we see that the equation x-\-k _ \ -{- fix + va^ ^ satisfied by the values a, h, c, d oi x\ and, therefore, since the equation in a? is ^^y of the third order, that the equation in question must be identicaUy true. We naay therefore write \ + fix-\-va^=^(px-^<T)(x + k\ \' + fix+v*a^ = (px + a)(x-\'^), *^^ the two systems of equations become therefore equivalent to the single system, V(a +p) (a + q) (a +/) (a -f q') = (pa + a) V(a + *)(« + *'), V(6+;?)(6 + 9)(6+;?')(6 + ?') = (p6 + <^) V(ft + A) (6 + A:'), V(c +p) (c + q) (c + Jt>0 (c + 9') = (pc + cr) V(c + A) (c + A:'), V(d +i>) (d + ?) (d +p') {d -h gO = (pd + cr) V(d + A:) (d + jfc'), 15—2 116 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [123 a set of equations which may be represented by the single equation where x is arbitrary; or what is the same thing, writing —a: instead of a?, Hence, putting J v(a? + a)(a: + 6)(a: + c)(a? + cO(a?-A:)(a?-A?')* J V(a?4-a)(a? + 6)(a? + c)(a? + c0(a?-A:)(a?-ifc') we see that the algebraical equations between p, q\ p\ <]( are equivalent to the transcendental equations Up ±liq ± Up' ± Ilgr' = const. TI,p ± U^q ± ny ±U,q' = const. The algebraical equations which connect 0, <f> with p, q; p\ ((, may be exhibited under several different forms; thus, for instance, considering the point (oo ; ^, </>) as a point in the line joining (A; p, q) and (A:'; p\ q% we must have V(a+p)(a + g) ^ VoTA, V(6 4-p)(6 + g') -^ \/6 + *,. . . = 0, i.e. the determinants formed by selecting any three of the four columns must vanish; the equations so obtained are equivalent (as they should be) to two independent equations. Or, again, by considering (oo ; d, ^) first as a point in the tangent plane at {k\ p, q) to the surface (k), and then as a point in the tangent plane at (Ar^; p\ q') to the surface (&'), we obtain 2 ((6- c)(c-d)(d - 6) V(a + p)(a + g) V(aTl) V(a + ^)(a4-<^)) = 0, 2((6-c)(c-(0(d-6)V(^TpOMY^ Or, again, we may consider the line joining (oo ; 6, <f>) and (k; p, q) or (A/; p\ q'), as touching the surfaces (k) and (A/); the formulae for this purpose are readily obtained by means of the lemma, — 123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. 117 " The condition in order that the line joining the points (f , i;, (^, ©) and (f', n\ ^, (o') inay touch the surface is 2ab(fi7'-r^)^ = 0. the summation extending to the binary combinations of a, b, c, dr But none of all these formulae appear readily to conduct to the transcendental equations connecting 0, with p, q; p\ q\ Reasoning from analogy, it would seem that there exist transcendental equations ±U0 ±U<f} tllp ± Up' =con8t. ± n^0 ± n,<f> ± U^p ± U^p' = const., or the similar equations containing q, ((, instead of j}, p\ into which these are changed by means of the transcendental equations between j), q, p\ ((. If in these equations we write ff^ (f/ instead of 0, ^, it would appear that the functions Up, lip', II^p, Tl^p' may be eliminated, and that we should obtain equations such as ±T10 ±U(I> ±U0' ± n<f)' = const. ± U,0 ± U^<f> ± n,^ ± n,</)' = const. to express the relations that must exist between the parameters 0, <(> and ^, <!>' of the extremities of a chord of the surface ic» + y» + ^ + t(;* = 0, b order that this chord may touch the two surfaces * (^ H- y* + ^ + 1^;") + our* H- 6y* + c^ + dt£;» = 0, Ar'(aj» + y» + ^ + «(;») -f aa;* H- 6y* H- c^ + dw" = 0. The quantities k, V, it will be noticed, enter into the radical of the integrals JIo?, II/p. This is a very striking difiference between the present theory and the analogous theory relating to conies, and leads, I think, to the inference that the theory of the polygon inscribed in a conic, amd the sides of which tovdt conies intersecting the conic in the same four points, cannot be extended to surfaces in such manner as one might be led to suppose from the extension to surfaces of the much simpler theory of the polygon inscribed in a conic, and the sides of which totich conies having double conta^ct with the conic, (See my paper "On the Homographic Transformation of a 8ur£Eu;e of the second order into itself," [122]). The preceding investigations are obviously very incomplete; but the connexion which they point out between the geometrical question and the Abelian integral involving the root of a function of the sixth order, may I think be of service in the theory of these integrals. 118 [124 124 ON A PEOPEETY OF THE CAUSTIC BY EEFEACTION OF THE CIECLE. [From the Philosophical MagcLzine, vol vi. (1853), pp. 427 — 431.] M. St Laurent has shown (Oergonne, vol. xviu. [1827] p. 1), that in certain cases the caustic by refiraction of a circle is identical with the caustic of reflexion of a circle (the reflecting circle and radiant point being, of course, properly chosen), and a very elegant demonstration of M. St Laurent's theorems is given by M. Gergonne in the same volume, p. 48. A similar method may be employed to demonstrate the more general theorem, that the same caustic by refraction of a circle may be considered as arising from six different systems of a radiant point, circle, and index of refr'action. The demonstration is obtained by means of the secondary caustic, which is (as is well known) an oval of Descartes. Such oval has three foci, any one of which may be taken for the radiant point: whichever be selected, there can always be found two corresponding circles and indices of refraction. The demonstration is as follows : — Let c be the radius of the refiucting circle, /a the index of refraction; and taking the centre of the circle as origin, let f, 17 be the coordinates of the radiant point, the secondary caustic is the envelope of the circle where a, /3 are parameters which vary subject to the condition the equation of the variable circle may be written {^2(a^ + y« + c«)-(f> + i7« + c')}-2(/i«a:-f)a-2(/i»y-i7)/9 = 0, 124] ON A PROPERTY OP THE CAUSTIC BY REFRACTION OP THE CIRCLE. 119 which is of the form the envelope is therefore Hence substituting, we have for the equation of the envelope, Le. for the secondary caustic, (A^^a^ + y' + c») - (P + 17' + c»)}» = 4c» {(Ai'a; - f )« + (M»y - 17)»}, which may also be written and this may perhaps be considered as the standard form. To show that this equation belongs to a Descartes' oval, suppose for greater con- venience 17 = 0, and write /Lt*(a5» + y"- c*) - p + c* = 2c/i V(a?-f)» + y» ; 1 . / IV multiplying this equation by 1 — 5, and adding to each side c^lfi ) 4.(3? — f)* + y«, we have (i-^.){M»(a^+y'-c)-?+c}+(«-f)'+y'+c(A*-iy = (a;-f)» + y» + 2c(/t*-i)V(x-f)» + y' + c»(M-y*; 0' Kdacing Bgun, multiplying the same equation by — (l — 5), and adding to each side we have ^' reducing, («_^' + y. = ||V(a,_f). + y. + |(l-Dp. 120 ON A PROPBRTY OP THE CAUSTIC BY REFRACTION OF THE CIRCLE, [124 Hence, extracting the square roots of each side of the equations thus found, we have the equation of the secondary caustic in either of the forms to which are to be joined Vi'-^'- ^^=f\/( .-'-M'j ^y c(.-y7(.-g.^.(- f.^y(. -^hy'' /fr fA.^\ .(|_^)V(«;-f). + y. Write successively, r=f . c' = c , /t*' = M , (1) A* / c («) ^■'\- (/9) r=f . "'-? • (7) c' = Cf "-T' («) (0 or, what is the same thing. f=r . c = c', ^=/*' . (1) («) c=-, 1 (/3) f=r, r (7) CSC*, (8) V 1:' > (Z ''=/• c' («) r=f . H' - =? A*" ^ ' H- r=f . 124] ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE CIRCLE. 121 or, again, (1) («) (7) (8) (*) then, whichever system of values of f, c', /Lt' be substituted for f, c, /a, we have in each case identically the same secondary caustic, the effect of the substitution being amply to interchange the different forms of the equation; and we have therefore identically the same caustic. By writing &c., ^ j9, 7, 5, € will be functional symbols, such as are treated of in my paper " On the Theory of Groups as depending on the symbolic equation 6^ = 1," [126], and it is ^ to verify the equations = ^8" = 87 = eS = 7€, /9s= a? = ey = 78 = Se, 7 = Sa = 6/9 = /98 = oe, S = ea = 7/8 = 07 = /Se, e = 7a = 8/9=/97 = aS. Suppose, for example, f= — c, i.e. let the radiant point be in the circumference; ™n in the fourth system ^ = — c, c' = — , (or, since d is the radius of a circle, this radius may be taken - ), /Lt' = — 1, or the new system is a reflecting sjrstem. This is ^^^ of M. St Laurent's theorems, viz. C. IL 16 122 ON A PROPBRTY 0*" THE CAUSTIC BY REFRACTION OF THE CIRCLE. [124 Theorem. The caustic by refiractiou of a circle when the radiant point is on the circumference, is the caustic by reflexion for the same radiant point, and a concentric circle the radius of which is the radids of the first circle divided by the index of re&action. Again, if f = — c/li, the fifth system gives ^ = ^^ , d — Cy /a' =^ — 1, or the new system is in this case also a reflecting system. This is the other of M. St Laurent's theorems, viz. : — Theorem. The caustic by refraction of a circle when the distance of the radiant point bom the centre is equal to the radius of the circle multiplied by the index of refiraction, is the caustic by reflexion of the same circle for a radiant point which Ls the image of the first radiant point. Of course it is to be understood that the image of a point means a point whose distance firom the centre = square of radius -r- distance. 2 Stom Buildings, Nov. 2, 1863. 125] 123 125. ON THE THEORY OF GROUPS, AS DEPENDING ON THE SYMBOLIC EQUATION ^=1. [From the Philosophical Magazine, vol. vii, (1854), pp. 40 — 47.] Let ^ be a symbol of operation, which may, if we please, have for its operand, not a single quantity x, but a sjrstem (x, y, ...)> ^^ ^^^^ 0(^, y, ...) = («^. y', ...), where a/, y', ... are any functions whatever of x, y, ..., it is not even necessary that x\ y\ ... should be the same in number with x, y, ..,. In particular a/, y', &c. may represent a permutation of x, y, &c., is in this case what is termed a substitution; and if^ instead of a set x, y, ..., the operand is a single quantity x, so that Ox^af =fx, fl is an ordinary functional symbol. It is not necessary (even if this could be done) to attach any meaning to a symbol such as tf ± <^, or to the symbol 0, nor con- sequently to an equation such as ^ = 0, or ^±^ = 0; but the symbol 1 will naturally denote an operation which (either generally or in regard to the particular operand) leaves the operand unaltered, and the equation = <l> will denote that the operation is (either generally or in regard to the particular operand) equivalent to kJ), and of course 0=1 will in like manner denote the equivalence of the operation to the operation 1. A sjnmbol 0<l> denotes the compound operation, the performance of which is equivalent to the performance, first of the operation (f>, and then of the operation 0; 0^ is of course in general different from ^^. But the symbols 0, 0, ... are in general such that ^ . ^ = ^0 . %> &c., so that 0^, 0<lyx^t ^^- ^^^^ ^ definite signi- fication independent of the particular mode of compounding the symbols; this will be the case even if the functional operations involved in the sjmcibols 0, ^, &c. contain pajrameters such as the quaternion imaginaries i, j, k; but not if these iiinctional operations contain parameters such as the imaginaries which enter into the theory of octaves, &c., and for which, e.g. a.l3y is something different from a^.7, 16—2 124 ON THE THEORY OF GROUPS, [125 a supposition which is altogether excluded from the present paper. The order of the factors of a product ^^;^... must of course be attended to, since even in the case of a product of two factors the order is material; it is very convenient to speak of the symbols 6, ^ . . . as the first or furthest, second, third, &c., and last or nearest factor. What precedes may be almost entirely summed up in the remark, that the distributive law has no application to the symbols ^^ ... ; and that these symbols are not in general convertible, but are associative. It is easy to see that ^=1, and that the index law ^.^=6?^+'*, holds for all positive or negative integer values, not excluding 0. It should be noticed also, that if tf = 0, then, whatever the symbols o, /8 may be, a0ff = a<f>0, and conversely. A set of symbols, 1, a, ^, ... all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a groups. It follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group; or what is the same thing, that if the symbols of the group are multi- plied together so as to form a table, thus: Farther factors 1 a /3 .. to a 13 1 a a )8« a* /9 • • «/3 /S* that as well each line as each column of the square will contain all the symbols 1, a, ^, — It also follows that the product of any number of the S3mibols, with or Avithout repetitions, and in any order whatever, is a symbol of the group. Suppost^ that the group 1, a, ^, ... contains n symbols, it may be shown that each of these sjrmbols satisfies the equation ^ = 1; so that a group may be considered as representing a system of roots of this symbolic binomial equation. It is, moreover, easy to show that if any symbol o of the group 1 The idea of a group as applied to permatations or substitutions is due to Oalois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraical equations. 125] AS DEPENDING ON THE SYMBOLIC EQUATION ^=1. 125 satisfies the equation ^ = 1, where r is less that n, then that r must be a sub- multiple of n; it follows that when n is a prime number, the group is of necessity of the form 1, a, a\...a^-\ (a«=l); and the same may be (but is not necessarily) the case, when n is a composite number. But whether n be prime or composite, the group, assumed to be of the form in question, is in every . respect analogous to the system of the roots of the ordinary binomial equation a:* — 1 = ; thus, when n is prime, all the roots (except the root 1) are prime roots; but when n is composite, there are only as many prime roots as there are numbers less than n and prime to it, &c. The distinction between the theory of the symbolic equation ^ = 1, and that of the ordinary equation a::* — 1 = 0, presents itself in the very simplest case, n = 4. For, consider the group which are a system of roots of the symbolic equation There is, it is clear, at least one root /3, such that /S" = 1 ; we may therefore represent the group thus, 1, a, /9, al3, (i8» = l); then multiplying each term by a as further factor, we have for the group 1, o*, afi, a'jS, so that a' must be equal either to or else to 1. In the former case the group is 1, a, a\ a», (a*=l), which is analogous to the system of roots of the ordinary equation ic* — 1 = 0. For the sake of comparison with what follows, I remark, that, representing the last- mentioned group by 1, a, A 7» we have the table 1, a, A 13 1 a 13 7 7 a 13 13 1 7 1 a /3 7 1 a 126 ON THE THEOllY OF GEOUP8, [125 If, on the other hand, ct* = l, then it is easy by similar reasoning to show that we must have a^ = /3cc, so that the group in the case is 1, a, /9, a^e, (a« = l, /3» = 1, a/8 = /3a); or .if we represent the group by we have the table 1 a i8 7 /9 1 a /9 7 a 1 7 7 1 a i 1 7 /9 a or, if we please, the symbols are such that a« = /3» = 7» = 1, /9 = 7« = «A 7 =a/9 = i8a; [and we have thus a group essentially distinct from that of the system of roots of the ordinary equation ic* — 1 = 0]. Systems of this form are of frequent occurrence in analysis, and it is only on account of their extreme simplicity that they have not been expressly remarked. For instance, in the theory of elliptic functions, if n be the parameter, and / \ ^ Q/ \ ^ + ^ / \ c»(l+n) then a, /8, y form a group of the species in question. So in the theory of quadratic forms, if a (a, 6, c) = (c, 6, a) 0(a, 6, c) = (a, -6, c) 7 (a, 6, c) = (c, - 6, a) ; although, indeed, in this case (treating forms which are properly equivalent as identical) we have a = /8, and therefore 7=1, in which point of view the group is simply a group of two symbols 1, «,(«'=!). 125] AS DEPENDING ON THE SYMBOLIC EQUATION ^ = 1. 127 Again, in the theory of matrices, if I denote the operation of inversion, and tr that of transposition, (I do not stop to explain the terms as the example may be passed over), we may write a = 7, i8 = tr, 7 = 7 . tr = tr . 7. I proceed to the case of a group of six symbols, 1, a, )8, 7, 8, e, which may be considered as representing a system of roots of- the symbolic equation It is in the first place to be shown that there is at least one root which is a prime root of ^ = 1, or (to use a simpler expression) a root having the index 3. It is clear that if there were a prime root, or root having the index 6, the square of this root would have the index 3, it is therefore only necessary to show that it is impossible that all the roots should have the index 2. This may be done by means of a theorem which I shall for the present assume, viz. that if among the roots of the symbolic equation ^ = 1, there are contained a system of roots of the symbolic equation 6^=1 (or, in other words, if among the symbols forming a group of the order there are contained symbols forming a group of the order j>), then j> is a submultiple of n. In the particular case in question, a group of the order 4 cannot form part of the group of the order 6. Suppose, then, that 7, h are two roots of ^=1, having each of them the index 2; then if 7S had also the index 2, we should have 7S = S7; and 1, 7, 8, S7, which is part of the group of the order 6, would be a group of the order 4. It is easy to see that 7S must have the index 3, and that the group is, in fact, 1, 78, S7, 7, 8, 7S7, which is, in fact, one of the groups to be presently obtained; I prefer commencing with the assumption of a root having the index 8. Suppose that a is such a root, the group must clearly be of the form 1, o, o*, 7, a7, a«7, (a»=l); and multipljring the entire group by 7 as nearer factor, it becomes 7, 07, a*7, 7-, ay, fltV; we must therefore have 7^ = 1, a, or a". But the supposition 7* = a' gives y* = a* = a, and the group is in this case 1, 7, 7*, 7*, 7*, 7" (7^=1); and the suppo- sition 7* = a gives also this same group. It only remains, therefore, to assume 7* = 1 ; ihen we must have either 7a = a7 or else 70 = 0*7. The former assumption leads to the group 1, a, a", 7, 07, a^ (a» = l, 7*= I, l^^^l\ which is, in fact, analogous to the system of roots of the ordinary equation afi—\ = ^\ and by putting 07 = X, might be exhibited in the form 1, \, V, X', \*, V, (\« = 1), under which this system has previously been considered. The latter assumption leads to the group 1, o, a«, 7, 07, o^ (a» = l, 7^ = 1, 7a = «^)> and we have thus two, and only two, essentially distinct forms of a group of six. If we represent the first of these two forms, viz. the group 1, «, a*, 7» «% a'Vi (a' = l, 7' = 1> 7a=«7) 128 ON THE THEORY OF OBOUPS, [125 by the general symbols we have the table 1. a. /9, 7, S, e, 1. a. /9, S, 1 a 1 7 S t* a fi 7 8 e 1 /9 7 S e 1 a 7 B e 1 a /9 ys b € 1 a 7 € 1 a P 7 S while if we represent the second of these two forms, viz. the group 1, a, a\ 7, 07, o^ (ct* = l, 7»=1, 70 = 0*7), by the same general symbols 1, a, /9, 7» S» €, we have the table 10/9786 1 /8 1 a ys 7 S e a /9 1 e 7 S y3 1 a S € 7 7 £ e 1 iS S € 7 ^ 1 a e 7 h 1 125] AS DEPENDING ON THE SYMBOLIC EQUATION $*'=l. 129 or, what is the same thing, the system of equations is 1 = ^a = 5r/9 = 7» =55=6*, a = /8^ = Sy = eS = ye, /8 = a' = €7 = yS = Se, fy = Set = e^ = ^g = a€, S = ea = y/8 = ay = /8e, £ = fya = S/8 = /9y = aS. An instance of a group of this kind is given by the permutation of three letters; the group 1, a, 0f % ^» f may represent a group of substitutions as follows : — abc, cahy bca, acb, cba, bac abc abc abc abc abc abc. Another singular instance is given by the optical theorem proved in my paper "On a property of the Caustic by refraction of a Circle, [124]." It is, I think, worth noticing, that if, instead of considering a, /3, &c. as symbols of operation, we consider them as quantities (or, to use a more abstract term, *cogi- tables') such as the quaternion imaginaries; the equations expressing the existence of the group are, in fact, the equations defining the meaning of the product of two complex quantities of the form w + aa + 6/8 4- . . . ; thus, in the system just considered, (wH-aa + 6/9 + cy + dS + C6)(w;' + a'a + 6'/8 + cy + d'S + e'6)= W + Aa + B/S + Cy + DS + Ee, where W = tuw' + a6' + a'6 + cc' + dd' H- ee\ A =wa' + iji/a+ bV + dc' + edf + cc', 5 = w6' + w/6 4- aa'-\- ed + cd' + de\ C =^wc' + w/c 4- da' + eV + 6d' + aefy D = wd! +v/d-{-ea' + cb' + ac' + be\ E =we' + w'e H- ca' + d6' + 6c' -^adf. It does not appear that there is in this system anjrthing analogous to the Baodulus <i;* + a^ + y* + -?', so important in the theory of quatemiona I hope shortly to resume the subject of the present paper, which is closely connected, not only with the theory of algebraical equations, but also with that of c. n. 17 130 ON THE THEORY OF GROUPS, &C. [125 the composition of quadratic forms, and the 'irregularity' in certain cases of the determinants of these forms. But I conclude for the present with the following two examples of groups of higher orders. The first of these is a group of eighteen, viz. 1, a, A 7, ayS, i8a, 07, 70, /97, 7/8, a/87, /97a, 70/8, a^a, ^87^ 70:7, a^rtP, ^7^a, where ct« = l, /9» = 1, 7» = 1, (/87)» = 1, (7«)' = 1, («^)' = 1, (a/87)» = l, (/37a)» = l, (7«/8)» = l; and the other a group of twenty-seven, viz. 1, a, Qi*, 7» y* 7«, «% 7«'» aY t"** ^T^i 7^. «V» 07a, 07*a, 0*70:, 0*7*0, 070*, 07*0*, o^ct*, o^a", 707', 70*7*, 7*07, 7*0*7, 7*070*, 707*0*, where o» = l, 7»=1, (7o)' = l, (yo)» = l, (7Qt«)' = l, (7'c^)'=l. It is hardly necessary to remark, that each of these groups is in reality perfectly symmetric, the omitted terms being, in virtue of the equations defining the nature of the symbols, identical with some of the terms of the group: thus, in the group of 18, the equations a? = l, /9* = 1, 7^ = 1, (0/87)* = 1 give 0^7 = 7/80, and similarly for all the other omitted term& It is easy to see that in the group of 18 the index of each term is 2 or else 3, while in the group of 27 the index of each term is 3. 2 SUyns Buildings, Nov. 2, 1853. 126] 131 126. ON THE THEOKY OF GKOUPS, AS DEPENDING ON THE SYMBOLIC EQUATION <9»=1.— Second Part. [From the PkUoaophical Magazine, vol. vii. (1854), pp. 408 — 409.] Imagine the symboU L, M, N, ... Bach that (JL being any symbol of the system), w the group 1, a, A---; then, in the first place, M being any other 83rmbol of the system, M^^L, M^^M, ^''%,.. will be the same group 1, a, /9, In fact, the system Z, M, N,.., may be ^tten Z, Za, L8...; and if e.g. if = Za, iV'=Z/9 then M-^N = (La)-' Z/9 = a-* Z"^ Z/9 = a-% which belongs to the group 1, a, A .... Next it may be shown that LL'\ ML-\ NL-\... ^ & group, although not in general the same group as 1, a, ^, .... In bet, writing -^^ia, N=Lfi, &c., the i^mbols just written down are LL''\ LolL-\ Z/9Z-S... *^d we have e.g. LoJr' . L/SL"' = LajSL"' = LyL''\ where 7 belongs to the group I, a, 0, 17—2 132 ON THE THEORY OF GROUPS, &C. [126 The system Z, M, JV, ... may be termed a group-holding system, or simply a holder; and with reference to the two groups to which it gives rise, may be said to hold on the nearer side the group Z~*Z, L~^M, Z""W, ..., and to hold on the further side the group ZZ~S LM"^, LN~^,.., Suppose that these groups are one and the same group 1, a, 13... y the system Z, Jf, -JT, ... is in this case termed a sym- metrical holder, and in reference to the last-mentioned group is said to hold such group symmetrically. It is evident that the symmetrical holder Z, M, N, ... may be expressed indiflferently and at pleasure in either of the two forms Z, Zo, Lj3,... and Z, ctZ, I3L', Le. we may say that the group is convertible with any symbol Z of the holder, and that the group operating upon, or operated upon by, a symbol Z of the holder, produces the holder. We may also say that the holder operated upon by, or operating upon, a symbol a of the group reproduces the holder. Suppose now that the group li a, A y> 8, €, f. ••• can be divided into a series of symmetrical holders of the smaller group 1, a, ^, ... ; the former group is said to be a multiple of the latter group, and the latter group to be a submultiple of the former group. Thus considering the two different forms of a group of six, and first the form 1, a, «•, 7» 7«» 7a'> (a* = l, 7* = !. «7=7a), the group of six is a multiple of the group of three, 1, a, o* (in fiwt, 1, a, o* and 7, 70, 7a* are each of them a symmetrical holder of the group 1, a, a'); and so in like manner the group of six is a multiple of the group of two, 1, 7 (in fact, 1, 7 and a, a7, and a, 0*7 are each a symmetrical holder of the group 1, 7). There would not, in a case such as the one in question, be any harm in speaking of the group of six as the product of the two groups 1, a, a' and 1, 7, but upon the whole it is, I think, better to dispense with the expression. Considering, secondly, the other form of a group of six, viz. 1, a, <^f 7» 7a, 7a"(a* = l, 7' = 1> a7 = 7a'); here the group of six is a multiple of the group of three, 1, a, a* (in fact, as be- fore, 1, a, a' and 7, 70, 70', are each a symmetrical holder of the group 1, o, o', since, as regards 7, 7a, 7a", we have (7, 70, 70') = 7(1, o, o«) = (l, o", 0)7). But the group of six is not a multiple of any group of two whatever; in feet, besides the group 1, 7 itself, there is not any symmetrical holder of this group 1, 7; and so, in like manner, with respect to the other groups of two, 1, 7a, and 1, 7a'. The group of three, 1, a, a\ is therefore, in the present case, the only submultiple of the group of six. It may be remarked, that if there be any number of symmetrical holdera of the same group, 1, a, )8, ... then any one of these holders bears to the aggregate of the holders a relation such as the submultiple of a group bears to such group; it is proper to notice that the aggregate of the holders is not of necessity itself a holder. 127] 133 127. ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE OF THE SECOND ORDER INTO ITSELF. [From the Philosophical Magazine, vol. vil (1854), pp. 208 — 212: continuation of 122.] I PASS to the improper transformation. Sir W. K Hamilton has given (in the note, p 723 of his Lectures on Quaternions [Dublin, 1853)] the following theorem : — If there l>e a polygon of 2m sides inscribed in a surface of the second order, and (2m - 1) of the sides pass through given points, then will the 2m-th side constantly touch two cones circumscribed about the surface of the second order. The relation between the extremities of the 2m-th side is that of two points connected by the general improper ^fansformation ; in other words, if there be on a surface of the second order two points such that the line joining them touches two cones circumscribed about the stir&ce of the second order, then the two points are as regards the transformation ^ question a pair of corresponding points, or simply a pair. But the relation between ^he two points of a pair may be expressed in a different and much more simple form. For greater clearness call the surface of the second order Z7, and the sections along which it is touched by the two cones, 0, <(>; the cones themselves may, it is <^W, be spoken of as the cones 0, <f>. And let the two points be P, Q. The line •PQ touches the two cones, it is therefore the line of intersection of the tangent plane through P to the cone 0, and the tangent plane through P to the cone 0. l^t one of the generating lines through P meet the section in the point A, and the other of the generating lines through P meet the section ^ in the point B. The tangent planes through P to the cones 0, if> respectively are nothing else than the tangent planes to the surface U at the points A, B respectively. We have there- fore at these points two generating lines meeting in the point P; the other two 134 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127 generating lines at the points A, B meet in like manner in the point Q. Thus P, Q are opposite angles of a skew quadrangle formed by four generating lines (or, what is the same thing, Ijdng upon the surface of the second order), and having its other two angles, one of them on the section and the other on the section ^ ; and if we consider the side PA as belonging determinately to one or the other of the two systems of generating lines, then when P is given, the corresponding point Q is, it is clear, completely determined. What precedes may be recapitulated in the statement, that in the improper transformation of a surface of the second order into itself, we have, as corresponding points, the opposite angles of a skew quadrangle lying upon the surface, and having the other two opposite angles upon given plane sections of the surface. I may add, that attending only to the sections through the points of intersection of 0, ^, if the point P be situate anywhere in one of these sections, the point Q will be always situate in the other of these sections, Le. the sections correspond to each other in pairs; in particular, the sections 0, <f> are corresponding sections, so also are the sections 6, <I> (each of them two generating lines) made by tangent planes of the surface. Any three pairs of sections form an involution ; the two sections which are the sibiconjugates of the involution are of course such, that, if the point P be situate in either of these sections, the corresponding point Q will be situate in the same section. It may be noticed that when the two sections 0, <f> coincide, the line joining the corresponding points passes through a fixed point, viz. the pole of the plane of the coincident sections; in fact the lines PQ and AB are in every case reciprocal polars, and in the present case the line AB lies in a fixed plane, viz. the plane of the coincident sections, the line PQ passes therefore through the pole of this plane. This agrees with the remarks made in the first part of the present paper. The analytical investigation in the case where the sur&ce of the second order is represented under the form xy — zw = is so simple, that it is, I think, w^orth while to reproduce it here, although for several reasons I prefer exhibiting the final result in relation to the form ic" + y* + ;?* + t(;* = of the equation of the surface of the second order. I consider then the surface xy--zw = 0, and I take (a, /9, 7, S), (a\ /S', y, S') for the coordinates of the poles of the two sections 0, ^, and also (^i> yi> ^i> ^i)> (^> ya» ^a» ^«) fts the coordinates of the points P, Q. We have of course a^i^i — j^iWi = 0, flPi^a — jSjWj = 0. The generating lines through P are obtained by com- bining the equation xy — zw = of the surface with the equation wi/i + yxi^zwi^wzi^O of the tangent plane at P. Eliminating a firom these equations, and replacing in the result Xi by its value -^-^, we have the equation (y^ - -^yi) (ywi - t(;yi) = 0. We may if we please take yzi — zyi^O, asy^^-yxi-- zWi^-wzi^O as the equations of the line PA ; this leads to yzi-zyi^O, I yw;«-w;y« = 0, | «y 1 + y ^ "" -8^1 "" ^^1 = , j wy^ + y^a — ^«^j — ^^^ = o, j ^a - zy^ = 0, I h + y^a — ^^a — t^^a = 0, j 127] OF THE SECOND ORDER INTO ITSELF. 135 for the equations of the lines FA^ QA respectively; and we have therefore the coordinates of. the point A^ coordinates which must satisfy the equation of the plane 0. This gives rise to the equation ya (ayi - S^i) - -J^a (7^1 - )9^i) = 0- We have in like manner ayi-¥yx^''ZWi-wzj = 0,] xy^ for the equations of the lines P£, QB respectively ; and we may thence find the coordinates of the point 5, coordinates which must satisfy the equation ^x + a'y - h'z - 7't(; = of the plane ^. This gives rise to the equation y% («Vi - 7^1) - z^ (8^1 - /S'wi)- It is easy, by means of these two equations and the equation x^^ — z^w^^O, to form the system a?a = (ayi - S^i) (a'yi - 7^i)> ya == (7yi - fiz^) i^'yi - ^^i)» w/a = (ayi - S-?,) (S'yi - fi'^^i) ; or, effecting the multiplications and replacing z^^Wx by x^y^y the values of x^y y,, 2^3, t^;. contain the common foctor yj, which may be rejected. Also introducing on the left- hand sides the common £Eu;tor MM\ where Jlf = a/9 — 78, M^ = a')8' — ^'i\ the equations become MM'x^ = 7'&c, + OLd'yi — a'S^^i — a7'«!i, MMy^ = /9i8'^+ 7S'yi - P^z^ - i8'7«'i, MMz^ = )87'iri + 7a'yi — /Sa'^Tj — 77'Wi , MMw^ = fflxx + aS'yi - SS'^j - dfi'w^, ^lues which give identically x^^ — z^^^ x^^ — z^w^. Moreover, by forming the value of the determinant, it is easy to verify that the transformation is in fact an im- proper, ona We have thus obtained the equations for the improper transformation of the surface xy^zw^^O into itself. By writing Xi + iyi, Xi^iyi for Xi, yi, &c., we have the following system of equations, in which (a, 6, c, d), (a\ b', &, d/) represent, as before, the coordinates of the poles of the plane sections, and Jlf' = a*H-5* + c' + d*, Jf'^«a'*H-6'«H-c'*H-(i^ viz. the system^ ^ The system b very similar in form to, but is euentially different from, that which conld be obtained ^ the theory of qnatemionB by writing the lasfc-mentioned transformatiGn is, in &ot, j^roper, and not improper. 136 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127 MWx^ = (oa' - hV - cc' - ddf) x^ + { ah' + a'b + cd' - c'd) yi + ( oc' + a'c + d6' - d'h) -g:, + ( ad'+ a'd + he - 6'c) w^, iOfy, = (a6' + a'6- cd' + c'd) iCi + (- oa' + 66' - cc/ - dd')y^ + ( 6c' + 6'c - da' + d'a) ^x + ( 6d' + 6'd - oc' + a'c) w, MM'z^ = (ac' + a'c - d6' + d'6) a^ + ( 6c' + 6'c '-'ad'+ a'd) yi -{-(-aa'- 66' + cc' - dd') ;?i + ( cd' -{- dd - 6a' + 6'a) w,, iOf'm, = (ad' + a'd- 6c' + 6'c) (x\-\-{ 6d' + 6'd -ca '\' da) y^ + ( cd' -{'dd- a6'+ a'6) z^ + (- aa' - 66' ''cd + dd') w^, values which of course satisfy identically x^-\-y^-\-z^-\-w^ = x^-\-y^-\-z^ + w^y and which belong to an improper transformation. We have thus obtained the improper trans- formation of the surface of the second order a^ + y' + 2:* + t(;* = into itself Returning for a moment to the equations which belong to the surface xy'^zw = 0, it is easy, to see that we may without loss of generality write a = ^=a' = )8'=0; the equations take then the very simple form MWx^^yBx,, MWy^^yB'yu M]irz^ = -yy'w,, itf Jf' w, = - SS'^j, where MJiT = V — 78 V — 7'S' ; and it thus becomes very easy to verify the geometrical interpretation of the formulae. It is necessary to remark, that, whenever the coordinates of the points Q are connected with the coordinates of the points B by means of the equations which belong to an improper transformation, the points P, Q have to each other the geometrical relation above mentioned, viz. there exist two plane sections 0, <l> such that P, Q are the opposite angles of a skew quadrangle upon the surface, and having the other two opposite angles in the sections 0, <(> respectively. Hence combining the theory with that of the proper transformation, we see that if A and B, B and C, ..., M and N are points corresponding to each other properly or improperly, then will JV and A be points corresponding to each other, viz. properly or improperly, according as the number of the improper pairs in the series A and B, B and C, ..., M and N is even or odd; i.e. if all the sides but one of a polygon satisfy the geometrical conditions in virtue of which their extremities are pairs of corresponding points, the remaining side will satisfy the geometrical condition in virtue of which its extremities will be a pair of corresponding points, the pair being proper or improper according to the rule just explained. I conclude with the remark, that we may by means of two plane sections of a surface of the second order obtain a proper transformation. For, if the generating lines through P meet the sections 0, <(> in the points A, B respectively, and the remaining generating lines through A, B respectively meet the sections ^, respec- tively in R, A', and the remaining generating lines through P', A' respectively meet in a point P'; then will P, P' be a pair of corresponding points in a proper trans- 127] OF THE SECOND ORDER INTO ITSELF, 137 formation. In fact, the generating lines through P meeting the sections 0, <f> m the points Af B respectively, and the remaining generating lines through A, B respectively meeting as before in the point Q, then P and Q will correspond to each other im- properly, and in like manner R and Q will correspond to each other improperly; i.e. P and P* will correspond to each other properly. The relation between P, P' may be expressed by saying that these points are opposite angles of the skew hexagon PARP'A'B lying upon the surface, and having the opposite angles A, A' in the section 0, and the opposite angles B, R in the section <(>. It is, however, clear from what precedes, that the points P, P' lie in a section passing through the points of intersection of 0, ^, and thus the proper transformation so obtained is not the general proper transformation. 2 Stone Buildings, January 11, 1854. c. n. 18 138 [128 128. DEVELOPMENTS ON THE PORISM OF THE IN-AND-CIRCUM- SCRIBED POLYGON. [From the Philosophical Magazine, vol. viL (1854), pp. 339 — 345.] I PROPOSE to develops some particular cases of the theorems given in my paper, "Correction of two Theorems relating to the Porism of the in-and-circumscribed Polygon" {Phil. Mag, voL vi. (1853), [116]). The two theorems are as follows: Theorem. The condition that there may be inscribed in the conic [7=0 an infinity of n-gons circumscribed about the conic F=0, depends upon the development in ascending powers of f of the square root of the discriminant of ^U+V\ viz. if this square root be then for n = 3, 5, 7, &c. respectively, the conditions are |C|=0, C, D =0, D, E G, D, E D. E. F E, F, = 0, &c ; and for n = 4, 6, 8, &c. respectively, the conditions are D|=0, D, E =0, E, F I D, E, F E, F, G F, G, H = 0, &c. 128] DEVELOPMENTS ON THE PORISM, &0. 139 Theorem. In the case where the conies are replaced by the two circles then the discriminant, the square root of which gives the series il + £f + Cp + Df + ^f * + &c. , is Write for a moment ^+£f+Cp + i)f> + ^f* + &c. = V(l + af)(l+6f)(l + cf), then ^ = 1. 25 = a + 6 + c, - 8C = a» + 6^ + c«- 26c - 2ca - 2a6, 162) = a» + 6» + c'-a* (6 +c)- 6^(0 + a)- c»(a +6) + 2a6c, -128i; = 5a* + 56* + 5c*-4a»(6 + c)-46»(c+a)-4c»(a + 6) + 4a»6c + 46»ca + 4c«a6 - 26»c» - 2c»a» - 2a>6«, &c. To adapt these to the case of the two circles, we have to write r-(l + af)(l + 6f)(l + cf) = (l+f){r»+f(r>+i?-a«) + f'i?l, and therefore c = l, values which after some reductions give ^=1. -r«.8C = (i? - a')" - 4i2V, r* . 162) = (i? - o») {(i? - o')' - 2r» (i? + a% -r* . 128^ = 5 (i? -o«)« - 8 (iJ»-o»)'(i? + 2r>)r»+ 16a«r«. Hence also »* . 1024 (GE - i)*) = {5 (i? - o«)* - 8 (i? - o»)' (ii' + 2r») r» + 16aV} {(i? - o>)* - 4i2V)} - 4 {(i? - o>)' - 2 (ii» - o») (i? + a') r»j', 18—2 140 DEVELOPMENTS ON THE POMSM OF THE [128 which after all reductions is + 16J? (i? + 2a») (i? - a»)» r* Hence the condition that there may be, inscribed in the circle a;' + y' — i? = and circumscribed about the circle' (a? — a)" + y* — r* = 0, an infinity of n-gons, is for n = 3, 4, 5, Le. in the case of a triangle, a quadrangle and a pentagon respectively, as follows. I. For the triangle, the relation is (i? - a»)» - 4iJV* = 0, which is the completely rationalized form (the simple power of a radius being of course analytically a radical) of the well-known equation which expresses the relation between the radii R, r of the circumscribed and inscribed circles, and the distance a between their centres. II. For the quadrangle, the relation is (i?-a»)»-2r»(J? + a») = 0, which may also be written (i2+r + a)(iJ + r-tt;(i2-r + a)(-B-r-a)-r* = 0. (Steiner, Crelle, t. ii. [1827] p. 289.) III. For the pentagon, the relation is (R''-'a''y - 12i? (i? -tt»)*r» + 16i?(i? + 2A^) (iP-a»)"r* - 64i?tt*r« = 0, which may also be written (iP - a^y {(i? - a^y - 4iPr»}» - 4iPr» {(iP - a^y - 4a»r»}» = 0. The equation may therefore be considered as the completely rationalized form of (i?-a*)> + 2iJ(i?-a»)»r-4i?(iP-a»)r»-8i2a»r«=a This is, in fact, the form given by Fuss in his memoir "De polygonis symme- trice irregularibus circulo simul inscriptis et circumscriptis," Nova Acta Petrop. t. xiii. [1802] pp. 166 — 189 (I quote from Jacobi's memoir, to be presently referred to). Fuss puts iJ + a=jt>, R — a = qy and he finds the equation jj^q^ - r' (/)» + y*) _ / g - r 128] IN-AND-CIRCUMSCRIBED POLYGON. 141 which, he remarks, is satisfied by r^^—p and r= -^ , and that consequently the rationalized equation will divide by p + r and pq — r{p-{-q); and he finds, after the division, P^^+p^^ip + q)r-pq(p + qyf^'-(p-\-q)(P'-qyr'=^0, which, restoring for p, q their values R + a, R — a, is the very equation above found. The form given by Steiner (CreUe, t. ii. p. 289) is r (iZ - a) = (iZ + a) V(i2 -T^M^) ~(^^r;^"^) + (22 + a^ which, putting p, q instead of iJ + a, -B — a, is qr =p 'Jip -r){q-r)-{-p V(g - r) (g +p) ; and Jacobi has shown in his memoir, "Anwendung der elliptischen Transcendenten u. 8. w.," CreUey t. III. [1828] p. 376, that the rationalized equation divides (like that of Fuss) by the factor pq-'ip + q)r, and becomes by that means identical with the rational equation given by Fuss. In the case of two concentric circles a = 0, and putting for greater simplicity . = Jlf, we have il+£f + Cp + Df> + ^f* + &c. = (l + f)Vl + Jff. This is, in fact, the very formula which corresponds to the general case of two conies having double contact For suppose that the polygon is inscribed in the conic (7=0, and circumscribed about the conic C7' + P* = 0, we have then to find the discriminant of (U+U + P*, Le. of (l + f)Cr+P*. Let K be the discriminant of U, and let F be what the polar reciprocal of U becomes when the variables are replaced by the coefficients of P, or, what is the same thing, let — ^ be the determinant obtained by bordering K (considered as a matrix) with the coefficients of P. The discriminant of (l+f)i7+P" is (1 + f)»ir + (l + f)»P, Le. it is (l+f)>{J5r(l+f) + P}, =(J5r + P)(l + f)>(l + iff), K ^here M== j^ — jj,; or, what is the same thing, M is the discriminant of U divided by the discriminant of U-^P*, And M having this meaning, the condition of there being inscribed in the conic (7 = an infinity of n-gons circumscribed about the conic ^ + P* = 0, is found by means of the series ^+5f+6'f»+i)p + ^f* + &c. = (l + f)Vl+iff. We have, therefore, DEVELOPMENTS ON THE P0RI8M OF THE A=l, 16/) = JP-2Jtf', ~12HE = 5M*-8M', 1024(C£-D') = i£*(AP-12if+16), Hence for the triangle, quadrangle and pentagon, the conditions are — I. For the triangle. [128 IL For the quadrangle, III. For the pentagon. if + 2 = 0. Jf- 4 = 0. and so on. It is worth noticing, that, in the case of two conica having a i-point contact, we have F = 0, and consequently M=l. The discriminant is therefore (l+f)*, and as this does not contain any variable parameter, the conies cannot be determined so that there may be for a given value of « (nor, indeed, for any value whatever of n) an infinity of n-gons inscribed in the one conic, and circumscribed about the other conic. The geometrical properties of a triangle, Sk. inscribed in a conic and circum- scribed about another conic, these two conies having double contact with each other. are at once obtained from those of the system in which the two conies are replaced 128] IN-AND-CIRCUMSCBIBED POLYGON. 143 by concentric circlea Thus, in the case of a triangle, if ABC be the triangle, and a, j8, 7 be the points of contact of the eidee with the inscribed conic, then the tangents to the circumscribed conic at A, B, C meet the opposite sides BC, CA, AB in points lying in the chord of contact, the linea Aa, Bff, Cy meet in the pole of contact, and BO on. In the case of a quadrangle, if ACEQ he the quadrangle, and b, d, f, h the pointa of contact with the inscribed conic, theu the tangents to the circumscribed conic at the pair of opposite angles A, E and the corresponding diagonal CQ, and in like manner the tangents at the pair of opposite angles' C, Q and the corresponding diagonal AB, meet in the chord of contact. Again, the pairs of opposite sides AC, £6, and the line dk joining the points of contact of the other two sides with the inscribed conic, and the pairs of opposite sides AO, GE, and the line hf joining the points of contact of the other two sides with the inscribed conic, meet in the chord of contact The diagonals AB, CQ, and the lines hf, dh through the points of coDtact of pairs of opposite sides with the inscribed conic, meet in the pole of ^tact, &c The beautiihl systems of 'focal relations' for regular polygons (in particular for tlw pentagon and the hexagon), given in Sir W, B. Hamilton's Lectures on Quaternions, [Dublin, 1853] Noa. 379 — 393, belong, it is clear, to polygons which are inscribed in and tuwmscribed about two conies having double contact with each other. In foot, the focus of s conic is a point such that the lines joining such point with the circular points at "ifinity (Le, the points in which a circle is intersected by the line infinity) are tangents ^ the conic In the case of two concentric circles, these are to be considered as ''"Jelling in the circular points at infinity; and consequently, when the concentric cucles are replaced by two conies having double contact, the circular points at infinity *K replaced by the points of contact of the two conies. 144 DEVELOPMENTS OK THE PORISM, &C. [128 Thus, in the figure (which is simply Sir W, E. Hamilton's figure 81 put into perspectiveX the system of relations F. G{..)ABCI, G. H{..)BCDK, B,I(..)CDEF, I, K(..)DEAG, K, F(. .) EABH. will mean, F, 0(.,)ABCI, that there is a conic inscribed in the quadrilateral ABCI such that the tangents to this conic through the points F and pass two and two through the points of contact of the circumscribed and the inscribed conies, and similarly for the other relations of the system. As the figure is drawn, the tangents in question are of course (aa the tangents through the foci in the case of the two concentric circles) imaginary, 2 Stone Buildivgs, March 7, 1854. 129] 145 I [ 129. ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE, AND ON AN IRRATIONAL TRANSFORMATION OF TWO TER- NARY QUADRATIC FORMS EACH INTO ITSELF. [From the Philosophical Magazine, vol. ix. (1855), pp. 513 — 517.] There is an irrational transformation of two ternary quadratic forms each into itself based upon the solution of the following geometrical problem, Given that the line Ix -\- my -\-nz = Dtteets the conic (a, 6, c, /, g, A$a?, y, zy = in the point (o^, y,, Zi); to find the other point of intersection. The solution is exceedingly simple. Take {a^, y^, z^) for the coordinates of the other point of intersection, we must have identically with respect to a?, y, z, (a, ...$a?, y, zy.(fSL, ...$i, m, wy-&(te + my + mr)» = (a,...$aH, yi, z^l^x, y, z).(a, ...Ja?,, y^, z^'^Xy y, z) ^ a constant foctor pris. Assume successively a?, y, ^r = ®, |^, ffi ; |^, 98, Jp ; ffi, ip, Ct ; it follows that C. n. 19 146 ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE, [129 or, what is the same thing, X.2 : y^ : Zt= yi^i (frw* + cfn^— 2/mn) ' ^1^1 (ctw' + &n* — 2hlm), It is not necessary for the present purpose, but it may be as well to give the corresponding solution of the problem : Given that one of the tangents through the point (f , 17, f) to the conic (a, 6, c, /, g, A$a?, y, zf = is the line fia? -f 7?i,y + ni^ = ; to find the equation to the other tangent. Let Ijc + m^ + n^^ = be the other tangent, then (a, ...$f, 17, t;y.{a,...\x, y, zf -[{a ...J^, 17, t$a?, y, z)Y to a constant factor prh. Assume successively y = 0, ^ = 0; £^ = 0, a? = 0; a: = 0, y = 0; then we have i, : m, : 71,= miWi {a (a, . . . $f , 17, ?)"-(af + Ai7+5rJ)»} : n,Z,{6(a,...$f, 17, ?)» - (Af + 617 +y?yi : iim,{c(a,...$f, 17, r)»-((7f+/^ + cO'}; or, as they may be more simply written, i, : 771, : n,= 7/^1^ (J8(r* + ®i7« + 2JP17O : 71, fi (ODp + ia?» - 2® tf ) Returning now to the solution of the first problem, I shall for the sake of simplicity consider the formulae obtained by taking for the equation of the conic, «^ + /Sy* + 7-8^ = 0. We see, therefore, that if this conic be intersected by the line lx-\-my + nz=^Q in the points {x^, y,, z^ and {x^, y„ j?,), then «j : yj : -^a = yiZi (77?i» + a/«») : ZiX^{an^ -\-fifi) 129] AND ON AN IRRATIONAL TRANSFORMATION &C. 147 We have, in fact, identically lyi^i (^w* + 7W1") + mZiXi (yP + cm*) + rw^i (am* + 0P) = {amnxi + firdyi + ylmzi) {Ix^ + myi + nz^) — J?/m {ax^ + /9yi' + 7-2^1'). «yi V (/9»" + 7*^")" + fiziW (yl* + an*)« + 7^1 V («^' + ^^')" = a/97 { - PiTi' — m'yi* — n'^i* + (twyi + nzi) Pa?i* + (n-2:, + Ix^) mh/^ + (Za^ + my^ n^y^ - Zlmnx^y^Zi] {Ix^ 4- wy, + /l^i) - (I'fiyxi" + w Vyi' + n^fiz,') (aci* + )9y,« + 7V) ; which show that if Ixi-^-myi^-nZi-O and aa?i» + )9yi* + 71^* = 0, then also lxi-^mya + nzi = and OiT,* + /Sya' + 7-er,* = : this is, of course, as it should be. I shall now consider Z, m, n as ffiven functions of Xi, yi, Zi satisfjdng identically the equations Ixi +myi +nzi =0, IHk + m^ca + n^b = 0, equations which express that lx + my + nz = is the tangent from the point (xi, y^, Zi) to the conic cur" + 6y* + cz^ = 0. And I shall take for a, /9, 7 the following values, viz. a = ax* + by I* + cz^* - a (a?i* 4- y^* + ^1*) , /9 = cuTi* + byi* + c-gTi* - 6 (x^* + y,* + ^1*), 7 = cur,' + 6yi' + c^i' - c (iTi* + yi* + -2^1*); 80 that iTi, yi, ^1 continuing absolutely indeterminate, we have identically ax^-k- ^yi^ + yz^ = ^. Also taking as a function of Xi, y^ ^,, the value of which will be subsequently pven, I write X2 = ©yi^i (fin* + 7m»), y2 = ^ZjXi(yP +an% Z2 = Sx^i(am*+l3t'); ^ that Xi, yi, ^1 are arbitrary, and x^, y^, z^ are taken to be determinate functions ^^ *i> yi, -2?!. The point (arj, yj, ^j) is geometrically connected with the point (xi, y^ z^) ^ follows, viz. (a?a, yj, ^,) is the point in which the tangent through (xi, yi, z{) to the conic cur" + 6y* + c^* = meets the conic passing through the point (a^, yi, ^Tj) and tie points of intersection of the conies ax* + b^ + cz* = and a:* + y* + ^' = 0. Con- ^uently, in the particular case in which (a?i, yi, ^1) is a point on the conic ^+y'+^ = 0, the point (x^, ya, -^2) is the point in which this conic is met by the t^gent through (a^, y,, Zi) to the conic ax^ + by* + cz* = 0, It has already been seen that Ix^ + myi + n^i = and aa?i^ + /9yi* + 7^1' = identically ; consequently we have identically Ix^ 4- my^ + w^Tj = and ouv^* + fiy^^ + 7^2' = 0. The latter equation, written under the form (ax^* + by,* + cz*) {xf + y,' + z^") - (iCi» + yi» + ^1') (cw?,- + by.^ + c^aO = «, 19—2 148 ON THE PORISM OF THE IN-AND-CIRCUM80RIBED TRIANGLE, [129 shows that if a^^ y„ z^ are such that a?j' + yj' + -?,• = a^' + y^' + ^j', then that also aoi^^ + by^ + cz^=^axi* + byi* + czi\ I proceed to determine O so that we may have ^i* + yi' + V = ^' + yi' + V« We obtain immediately ^,(^«' + y«' + '^>') = (^V + mV + »V)(aV + i8»yi« + 7«^i«) write for a moment CM?i* + 6yi' + C2r,'=p, iTi' + yi' + -2^1* = J, so that a=*p^aq, fi^p — bq, y^^p — cq, then o»ah' + I3h/,^ + T*^,' = 2p* - 2p .;)5 + (aV + 6»yi« + (fz^*) (f,^q {(aV + ^V + c»^i') 3 -i>*}. = J {(6 - c)« yiV + (c - a)» 2r,V + (a - 6)' fla'y,'}, a?iV + i8*wiV + Y*»iV - 2fiymWyi%* - 2yan^l^zW - 2a/8i»m'j?i«yi« — 2pg {oZ^ar/ + fcm^y/ + c?i*V — (6 + c) rnWy^z^ - {c + a) nH^z^x^ — (a + 6) 1hii^x^y^\ the first line of which vanishes in virtue of the equation Ix^ + myx + w^i = ; we have therefore ^(a^'+y,' + -?,•) -H (^1* + yi» + z^^) = (Z V + ^V + wV) {(6 - c)» yi V + (c - a)» Vj?i* + (a - 6)» iCj V} + 2 (aa?i* + 6yi' + cz^^) {aZV + &^V + cn%^-(b + c) mWy^z^- (c + a) nH%W - (a + 6) ?mV,*yi»} - (^' + yi' + -8^1*) {a'^V + 6*m^/ + c^V - 26cm«n«yi V - ican^l^z^x^ - iabl^Wyi^]- Hence reducing the function on the right-hand side, and putting we have + (c*wi* - 26«m V) yi V + (cM - 2c»n*P) -^i V + ( W* - 2a«Pm«) aj^ V + (bW - 2c*mV) yiV + (c»Z* - 2aWZ«) ^^ V + (aHn* - 26»Z>m«) iCj^y/ + ZmW {bc-ca''ab) + 2n'Z' (-6c + ca-a6) + 2Pm* ( - 6c - ca + a6)} Xi%%\ The value of might probably be expressed in a more simple form by means of the equations Ixi + myi + n^i = and Z'6c + m^ca + n*a6 = 0, even without solving these equations; but this I shall not at present inquire into. 129] AND ON AN IRRATIONAL TRANSFORMATION &C. 149 Recapitulating^ I, m, n are considered as functions of ^, y^, ^i determined (to a common £EU^tor pria) by the equations i*a + wiyi +nzi =0, Pbc + rn^ca + n^ab = ; B is determined as above, and then writing we have y, = ©^iri(7p +an'), ^, =5 ©a?iyi (am* + fiV) ; and these values give Is^ +my, + n2r, = 0, xf +y,' +-?,» =a?i« +yi' +zx\ axf + tyt* + ozf = cuTi' + 6yi' + c^i*. In connexion with the subject I may add the following transformation, viz. if Z^Jaaf^ V3/8 (y - -f) + V(3a - 2/8) (dj» + y» + z") + 2/8 (y;? + ;?a? + «y), f then reciprocally 3>//8a?=r-V3a(y'-/) + V(3/8-2a)(a/» + y'» + /«) + 2a(y'/ + /«^ + fl?'yO. a^ + y» + ^ =a?'» + y'« + /«, /8 (aj* + y* + jt* -y-z^ - -^a? - icy) = a (a^» + y'* + /' -y'-^' - -2'a?' - a?'yO. Suppose 1 + /:> + p^ = 0, then a^ + j^-{-2^ — yz'-zx — xy = (x + py + p^z) (a? + p* + p^) ; and in feet 3Va(a?' + py'+p»/) = ->/3)8(l + 2p)(a? + py + p«^), 3Va(a?'+py+p^) = V3i8(l + 2p)(a? + p»y + p^). The preceding investigations have been in my possession for about eighteen months. 2. Stone Buildings, April 18, 1855. 150 [130 130. DEUXIEME MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [From the Journal de Maihimatiques Pures et Appliques (Liouville), torn. xix. (1854), pp. 193—208: Sequel to Memoir t. x. (1845), 35.] Je vais essayer de d^velopper ici les propri^t^s qui se rapportent aux transformations lin^aires des p^riodes des fonctions yw, gx, Gx, Zx, dont je me suis occupy dans le M^moire sur les fonctions doublement p^riodiques que j*ai donn^ dans ce Recueil en 1845. Avant d*entrer en matifere, je remai^que que partant des expressions des deux p^riodes, oi i = V — 1, on obtient, en ^crivant fl* = ck) — a)'i, T»=i;-i/i, les Equations ft*T = wu + G) V + i (W - w'u), au moyen desquelles et des valeurs IIT mod. {(OV - do'v) ' SIT mod. {mv - (o'u) des quantity fi, B, on ddduitles formules ft mod. {(OV — a)'i;) ' T mod. («t;' — (o'v) ' 130] DEUXltlfB ICl^OIRE SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 151 Je ne &is attention qu'aux transformations qui correspondent k des entiers impairs et premiers, et je suppose, de plus, que la transformation soit toujours propre et r^gulifere ; c'est-i-dire qu'en ^rivant (2* + 1) ft, = xn + mT = (X, /Lt), (2A + l)T, = i;ft + pT = (i/, p), oh 2k +1 est un entier positif, impair et premier, et oh X, fi, v, p sont des entiers tels, qu'au signe pr^ \p-^fiv soit 6gsA k 2A; + 1, je suppose Xp — fiv=2k + 1, (condition pour que la transformation soit propre), et, en outre, X= 1, /iA = 0, (mod. 2) 1^ = 0, p=l, (condition pour que la transformation soit r^guli^re). On trouve tout de suite ft= pn,-/tT, = (p, -/^X, T=-.|/n, + XT, = (-i., XX; j'^cris aussi ft, « tt), + ©/», ft,* = tt), - «/», et je suppose que i3„ /3, soient des fonctions de o>,, v, telles que les fonctions B, 0, de 0, V. Cela ^tant, je forme d'abord T^uation (2* + 1) (o),v/ - o)>,) = 0)1/' - w't;, au moyen de laquelle T^uation ^ ^^ mod. (a)i/ - G)'i;) 8e transforme en Beli i<"^')(^^^>-^-i"- „,.d(.;--.>.) {°--°^=nf^}' mod. mod. 9rt(<»,v/-o)>,) ~ nX mod («,v/ - «>,) ^"' "^ '* * "' • 152 DEUXlklCB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^RXODIQUES. [130 oa enfin et de mSme {2k + l(B-fi)-B,]a^^fi^(p, X); Equations qui seront bientdt utiles. Je suppose d'abord que 2k + 1 soit ^gal k Tunit^, transformation que Ton peut nommer trivicUe, La fonction yx est d^finie par I'^quation ya? = e-**^a?n|l + ^^^J, mod.(m, n)< T, r= oo; dans (m, n) = mCl + nT, les entiers m, n doivent prendre toutes les valeurs positives ou negatives (le seul syst^me m = 0, n = except^) qui satisfont k Tin^galit^ mod. (m, n) < T, dont le second membre T sera ensuite suppose infini. Soit y,x la fonction corres- pondante pour les p^riodes fl^ , T^ ; on aura V ar = €-**.«• aril |l +7-^.1, mod.(m, ny<r, r=«. Or = m (Xil + fiT) + n (i/fl + pT), = (Xm + vn)il + (jjLfn + pn) T, = m^fl + n,T, En ^rivant, comme nous venons de le faire, m, = Xm + im, on voit tout do suite qu'ii chaque systfeme de valeurs entiferes de m, n, correspond un Hystfemo, ct un seul systfeme, de valeurs enti^res de m^, w,; et que de m6me k chaque syst^me de valeurs entiferes de m,, n,, correspond un systfeme, et un seul syst^me, do valeurs untiferes do m, w ; de plus, les systfemes m = 0, n = et m, = 0, n^ = 0, correspondent Tun k Tautre. II est done permis d'^rire ( (m, n)j ( (m, n),) les limitoH comme auparavant ; car, k cause de (m, n), = (m„ n,% gx 130] DEUXrfeMB Ml^OIRE SUR LES FONCTIONS DOUBLEMENT PiaaiODIQUBS. 153 la condition pour les limites, savoir : mod (m, n\ < T, T= oo , devient mod. (m, n) < r, T = oo . Cela donne enfin I'^uation et, au moyen de cette Equation, on obtient une ^nation correspondante pour la trans- formation de Tune quelconque des fonctions yx^ gx, Ox, Zx, d^finies par les ^nations yar = 6-**^.a;n|l+^^^^|, mod. (m,n)<r, = e-**^. nil 4-^-^^}, mod. (m, n) < T, r=oo; Ox^e^^. n(l+7-^J, mod. (m, fi) < r, \ (m, n)j Za? = €-**^. u\l+'-J-X, mod. (m, n) < r, (^nations dans lesquelles m = m + ^, n = n + ^). Je prends par exemple la fonction gx, et j'&ris dans I'^uation entre y^x et yx, x + ^fl au lieu de x. Soit pour un moment p = 2p' 4- 1, /A = 2fi\ cela donne a?+jn = ar + i(/)ft,-A*T,)=a? + (p',-/). Done y, (x + ill) = e^.*'P. '*')' M^g^x, c'est-i-dire y, (a? + ifl) = 6*^'*<^» **>' if,g,aj ; de plus, y (x + ill) = 6*^0* ifgar. Ces substitutions ^tant eflFectu(?es, les coefficients M, M, doivent 6tre ^limin^s en ^rivant ««0; cela donne ou enfin, au moyen d'une ^nation ddji trouv^, et de m^me pour les fonctions Ox, Zx, c. n. 20 154 DEUXlilME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. [130 Done eofin, en reprdsentant par Jx Tune quelconque des fonctions jx, gx, Ox, Zx, on aura oil J,x est ce que devient Jx au moyen d'une transformation triviale (propre ct r^gulifere) des p^riodes. Je passe k pr^nt k la transformation pour un nombre impair et premier {2k + 1) quelconque; mais pour cela on a besoin de connattre la valeur de la fonction "' = "{^-^ (m, r)+y }' ^od.[(m.n) + y}<T, r=oc. o{i y = a + bi est une quantity r^elle ou imaginaire quelconque. Soit u ce que devient u' en prenant pour la condition par rapport aux limites mod. (rw, n) < T, 7= oo ; on trouve sans peine y(y) Pour trouver u\ je forme T^uation u : u' = u\l 4- . ^ J , la limite inf^rieure du produit infini double ^tant mod. {(m, n) + y]> T, et la limite sup^rieure mod. (m, n) < T, T= x ; cela donne logtt-log«'=..S(-^;i7^-i^S,(^, „\+y}.+ -. car on peut d^montrer que S 7-^=0. ^7-^=0, &c. ■^^ (ot, n)* (m, n)* Pour cela, observons que m et n ^tant infinis puisque T Test, la premifere des sommes dont il s'agit peut se remplacer par rint^grale double J __ rCdmdn 130] DSUXI^ME M]^OIB£ SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 155 laquelle (en Ajrivant m = r cos d, n = r sin ^, ce qui donne, comme on sait, dmdn = rdrdff) devient -// drd0 r{ilcos0-{'r8ui0y' rfoti (log r) dd -II (aco&e+rQindy en prenant (logr) entre les limites convenablea Pour trouver ces limites, j'^ris (m, n) + y = r(ncos^ + TsiD^) + y; ce qui donne mod.' {(m, n) + y} = {r (flcos ^ + T sin ^) + y} {r(fl* cos d + T*8in 0) + y*}, fiavoir, k Tune des limites r»(ftcos^ + T8ind)(n*co8^ + T*8in^) + r{y*(nco8^ + T8in^) + y(ft*cos^ + T*sin^)} + 2* = 0; ou, en n^gligeant les puissances negatives de T, T r = V(n cos -h T sin 0) (ft* cos ^ + T* sin 0) 1 f y y* I "*t"cos^ + T8in^"*"ft*cos^ + T*8in^j' et it Tautre limite, r = V(ft cos e + T sin 0) (ft* cos ^ + T*sin ^) * Or, en repi^sentant ces deux ^nations par ?• = 22 — <^, r = 22, on trouve, pour la valeur de (log r) entre les deux limites, logi2-log(i2-^) = -log(l-|) = 0, i cause de la valeur infinie de 12. Ainsi la somme cherch6e est nuUe ; et il est tout clair que les sommes suivantes S / ^ » &c., se r^uisent de m6me k z^ro. (m, n/ Done enfin, 20—2 156 DEUXifeME MilMOIRE SUR LES FONCTIONS DOUBLEMENT PilRIODIQUES. [130 Cela fait voir que u' = e"** M, le coefficient k ^tant donn^ au moyen de T^uation o{i la somme est prise, comme auparavant, entre les limites mod {(m, n) + y} > T, mod. (m, n) < T, r=oo. Mais il n'est pas permis d'^rire J J (m, n) En eflTet, cette intdgrale n'est que le premier terme d'une suite dont il faudrait, pour obtenir un r^ultat exact, prendre deux termes; le second terme de la suite serait une int^grale prise le long d'un contour, et il serait, ce me semble, tres-difficile d en trouver la valeur. Pour trouver la valeur de A, je remarque que k sera fonction lin^aire des quantity T^ y, y*, 4f , &c., qui entrent dans les valeurs de r ; done, puisqu'en demifere analyse I'=oo, k ne pent 6tre que de la forme Ly -{- My*, Cela ^tant, en substituant pour u' sa valeur, je forme T^uation y(^+y) _ g_j«,t g(-.si/+Ly+iry)« n II + - 1 y(y) * I {rn,n) + yy mod. {(m, n) + y]<T, r= oo , et j'&ris successivement ce qui donne pour les valeurs correspondantes du produit infini double e"**** . gx et €~****. Ox ; en comparant les valeurs ainsi obtenues avec les ^nations qui donnent les valeurs de y(a? + in), y(af + iT), on trouve 2i = 0, M^ '^ mod. («i/ — oi't;) * ou enfin, y(a: + y) ^ ^_^^ /-^^mod. (l->-v)^)^ n(l+— ^— I y(y) ' 1 Kw) + yJ* mod. {(m, n) + y} < T, r= oo , laquelle est I'^uation qu'il s'agissait d'^tablir. II est & peine n^essaire de faire la X remarque que pour y = 0, on doit consid^rer & part le fiawteur 1 + - , lequel multipli^ if par y(y) devient tout simplement x\ T^uation subsiste done dans ce cas. 130] DEUXIjkaiE Ml^OIBE 8UR LES FONCTIONS DOUBLEMENT PilRIODIQUES. 157 En revenant au probl^me des transformations lindaires, partant des ^nations (2*; + l)ft, = Xfl+/AT, (2A: + 1)T, = i/n+/)T, je suppose d'abord que les coefficients X, v ne satisfassent pas k la fois aux deux conditions X = 0, 1/ = 0, mod. (2*; + 1), etje prends p, q des entiers quelconques tels, que \p-\-pq ne soit pas =0, mod. (2A; + 1). Cela ^tant, soient \p + vq =;>„ fMp + pq =gr„ (2*: + l)t=i>/"+3.T, et, par cons^uent, Je forme T^uation savour c'estri-dire \m + im — «p, = (2k -f 1) m^, fjLm + i/n — sq, = (2A; + 1) n^ , ou, ce qui est la m^me chose, n ^ sq = m^ fjL -- n^X. Or, m^, n^, ^ ^tant des entiers donn^, m, n seront aussi des entiers; de m6me, m, n ^tant des entiers donnas, on trouve de A: i — i un entier a qui donne m^ un entier. V&is cela ^tant, n, sera aussi un entier; car autrement n^ serait une fraction ayant pour d^nominateur, lequel on voudrait, des nombres 2& + 1, X, i/, ce qui est impossible i moins que X = 0, V = 0, mod (2A; + 1). ^^ si ces ^nations avaient lieu, on trouverait d'abord 8 de mani^re h, avoir n, entier, ^t alors, puisqu'on n'a pas aussi /A = 0, p = 0, mod. (2* + 1) (^u effet, cela est impossible k cause de T^uation Xp — /ty = 2i + 1), on d^montrerait, «>nmie auparavant, pour n^, que m, est entier. Done, enfin, rw, n ^tant des entiers doim&, on trouve pour m,, n,, « un systfeme d'entiers tel que 8 soit compris de A: it — A, «t Von voit sans peine qu'il n'y a qu un seul systfeme de cette espfece. 158 DEUXltME MISmOIBE SUB LES F0NCTI0N8 DOUBLEMENT pArIODIQUBS. [130 A pr&ent, partant de T^uation y (y) I K. w,) + yj (et faisant attention k la particularity que prints le cas de y = 0), j'^ris successivement y-0. y = ±-<^,..., y = ±h^, et je forme le produit des ^nations ainsi trouv^e& Cela donne, k cause de (tn,, n,) + «^ = (m, n),, y(»f) I (»». «)/J la condition, par rapport aux limites, ^tant mod.(m, n),<T, T=ao. Or avec la m^me condition, par rapport smx limites; done, enfin, y^,.,-*t.,-^.«^..n|y^>|, oil, dans le num^rateur, s doit avoir toutes les valours enti^res depuis s^ — k jusqu'i « = + A:, y compris « = 0, et dans le d^nominateur ces m6mes valeurs, hormis la valeur 8 = 0. II est, k present, facile de faire voir que cette propri^t^ subsiste pour I'une quelconque des fonctions yx, gx, Ox, Zx\ en eflTet, pour la d^montrer pour gx, jecris X + ^11 au lieu de a? ; en prenant, pour un moment, p = 2p' + 1, /a = 2/a', cela donne c*est-i-dire y, (*fi) Or, on d^uit de I'expression pour y^x, ■ y,(ifi) y>+Jii) = eW.«(P.M),g^a,. = e-tf ,» (P. M) , w (B,-5+u) (*»+ite) n y (^ + *^ + i") = e-V.<* (p. M), £-4 (»,-5+i«)ir« ji dis+i) (Mte ij g (^ + ^Y*) . 130] DBUXltME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. 159 ou enfin, k cause de T^uation la valeur de g^x est a a; = €-* (*.-^+^^«« . n g (^ + y . »' g («V^) et en repi^sentant, comme auparavant, Tune queleonque des fonctions yx, gx, Gx, Zx par Jx, on a T^uation equation dans laquelle 8 doit avoir, dans le num^rateur, t-outes les valeurs entiferes depuis « = — i jusqu'i 8 = k, y compris « = 0, et dans le d^nominateur, ees mSmes valeurs, hormis la valeur « = 0. Je suppose que les valeurs de p„ q, soient donn^es (cela va sans dire que Ton ne doit pas avoir k la fois p^sO, j, = 0, mod.2A: + l), et je remarque que Ton a, pour determiner X, fi, v, p, les conditions pp, - vq^ = 0, mod. {2k + 1), X=l, /A=0, mod. 2, v=0, p=l, Xp — fiv = 2k-\- 1. ^^ cela ^tant, on aura ensuite, en rassemblant toutes les Equations qui ont rapport k la transformation, Pi>/ -»'?/ = (2* +!);>, (2A + l)n, = Xft +mT, (2A + l)T,= i/ft +pT. 160 DEUXltME liiafOIRE SUR LES FONCTIONS DOUBLEMENT P^UUODIQUES. [130 Or, quoique les valeurs de X, /i, v, p ne soient pas compl^tement d^termin^es au moyen de ces conditions, cependant il est clair que la valeur de la fonction J^x ne depend que des valeurs de p„ q, (en eflfet, ces valeurs suffisent pour determiner la quantity '4^=p^fl + gr^T, de laquelle depend la fonction J^x). Les formes diflKrentes de J,x, pour les systfemes de valeurs de X, ft, v, />, qui correspondent k des valeurs donn^es de p,, q,, doivent done se d^river de Tune quelconque de ces formes, au moyen dune transformation triviale des modules 11^, T^. II est, de plus, clair que les valeurs de p,, q,, qui sont ^gales k des multiples de (2A; + 1) pr^, ne donnent qu'une seule valeur de J,x. Je suppose d'abord que p, = 0, mod (2* + 1), on pent trouver un entier tel que 0p,^l, mod.(2A: + l); en prenant alors 0q, = q,, mod(2A? + l), cela donne (p,a + q,T,) = ft + qX mod. (2* +1), savoir ^ = ft + qX Mttod. (2*; + 1). Mais en donnant k 8 des valeurs entiferes quelconques, depuis — k jusqu'& k, le syst^me des valeurs de syjt est equivalent au syst^me des valeurs de 80^, mod.(2A;+l); il est done permis d'^crire, sans perte de g^n^ralite, ^ = ft + g,T. De m^me pour p, = 0, mod. (2k + 1), on d^montre que Ton pent donner k q, une valeur quelconque, sans changer pour cela la valeur de J^x\ il convient d'avoir p^ impair et q^ pair. J'^cris done, pour le premier cas, 2q^ au lieu de g^, et je suppose que, dans le deuxi^me cas, les valeurs de p,, q^ soient p, = 2A: + l, 5, = 2. Cela donne: Premier ca8. ^ = ft + 2q,T, q^ un entier quelconque, y compris z^ro, depuis —A? jusqu'^ +A:. DeuxUme ca8. ^ = (2ifc + l)ft + 2T; le nombre des valeurs diff^rentes de "9 sera done, en tout, 2A; + 2. 130] DKUXliOIB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^BIODIQITES. 161 On obtient tout de suite, pour le premier cas, le syst^me d'^uations X = l, /*= 2q„ p = l, 5 = 0; ^ = 2A+-l(" + 2q,T). Le cas particulier le plus simple est eelui de 9^ = 0; cela donne ^ = "' = 2lTl"' '^' = '^' et, de 1^, 11 1 n T, 2ifc + l T' et m^me le cas g^ndral se r^uit k celui-ci, car, au moyen d'une transformation irimale, on obtiendrait ft' = n + 2q,T, T' = T. et puis Vr = ft,= 2k + l ft', T, = r, et, de ]&, 1 ft' ft T, 2k + lT' Les ^nations correspondantes pour le deuxi^me cas sont: p, = 2k+l, 9, = 2, \=2A+1, /t=0, V =0, p = 1, p=l. 9=2, ^ = 2*1:1 t^^* "^ ^^ " "^ ^^ ft, = ft. T = ' 2k + l T; ^ qui donne c. n. 21 162 DEUXltME MJ^OIBE SUB LES FONCTIONS DOUBLEMENT PI^ODIQUES. [130 J'ajoute, sans m'arrSter pour les d^montrer, quelques formules de transformation pour le nombre 2; je trouve d'abord Ces ^nations donnent, en introduisant les fonctions elliptiques, <f)x, fx, Fx donn^ au moyen de , yx - gx „ Gx ^=fc' >=k' ^^=:^' les ^uations F,x ^ Fx ' dont la seconde peut encore s'^rire sous la forme et les deux ^nations combin^es ensemble conduisent sans peine & la valeur des modules c^, 6^. On trouve en effet, en mettant comme & Tordinaire 6' = c' + 6*, c/ = 46c, 6/ = (6-c)», et puis - 1 — c (c — 6) i?x *'^= Wa; ' - _ 1 - c (c -h 6) ^'a ? •^'^■■l-c(c-6)<^«a?' F.x=^ l-c(c-6)^fl?' formules qui correspondent & celles de la transformation de Lagrange. Les ^nations pour y^a?, Z^x donnent encore une valeur de ^^a?, laquelle, ^gal^ k la valeur qui vient d'etre trouvfe, donne yx%xZ'^(\iX) _ if>xfx Z{x''\il)Z{x + Jft) " 1 - c(c-6) <^«a? * 130] DEUXliaCE M^OIRE SUB L£S FONCTIONS DOUBLEMENT P^BIODIQUES. 163 On obtient tout de suite les formules pour la transformation analogue il, = il, T^ = ^T. Mais il &ut de plus consid^rer le syst^me on aura alors et puis, en Anrivant on obtient ll, = i(n-T), T, = i(n + T): r yx = €~* <*'-*^^ gX Zx, -^_,-...-«.^y (^+in-T)y(5-4n-T) ^' -yan-t) z* ( jn - T) -y»(in + T) ^» (in + T) c,» = (c - tc)«, - e; = (e+ icy, Ax ^'* ^fiTF'x ' , 1 +ice4^x J'" ~ fxFx ' l-tce4fx l + tce<ya; Z(a; + in-T)Z(a;-ifl-T) fx Fx Z* (iir^) gx Ox l-ice^fx Z(a! + ifl + T)Z(a;-^ + T) fxFx Z*{^ + t)gxOx \-\-iee^x ^ Z{x + \€l-T)Z{x-^-T)Z>(^ + r) ^ \-ice4fx Z {x + ^oTT) Z (x - ^T^t) Z* {^ -T)' 01, an moins, ces formules seront exactes au signe de i prfes; car il serait peut-6tre *"ffi<ale de determiner quel est le signe qu'on doit domier h, cette quantity. 21—2 164 [131 131. NOUVELLES RECHERCHES SUR LES COVARIANTS. [From the Journal filr die reine und angewandte McUhematik (Crelle), torn. XLvn. (1854), pp. 109—125.] Je me sers de la notation (oo, ai,...an)(a?, yT ' pour repr^enter la fonction en supposant que lea coefficients ao\ a^ &c. soient donnds par T^quation (oo, Oi, . . . On) (X^ + AAy, \'x + fiyY = (oo', Oi', . . . O (a?, y)\ suppose identique par rapport k x, y, soit ^(oo, ai,...an\ x, y) une fonction des co- efficients et des variables, telle que ^K', Oi', ...On'; X, y) = (X/-X»P^(ao, ai,...an; \x + fiy, \'x'\-fify)] cette fonction ^ sera gdn^ralement un Covariantt et dans le cas particulier oil if> est fonction dea seuls coefficients, un Invariant de la fonction donn^. Je suppose d'abord que les nouveaux coefficients soient donn^ par T^uation . (Oo, Oi, ... an)(x + \y, y)~ = (ao', a,', ... an')(x, y)«; cela donne les relations Oi' = Gti + XOo, Oa' = ttj + 2\ai + X'Oo, &c. 131] NOUVELLES BEGHERCHES SUB LES COVABIANTS. 165 n £Biut done que le covaria/nt ^ satisfasse & T^uation 4>W> <^> ... On'; a?, y) = <^(ao, Oi, ... a„ ; a? + Xy, y), laquelle peut aussi Stre ^rite comme suit: 4>W> ai',...a„'; a?-Xy, y) = 0(ao, Oi, ...On; a:, y). (Z) De mSme, en fiEusant (Oo, Oi, ...a„)(a?, fix + yy^ioo", (h\...an")(x, y)«, ce qui donne On' =an le cooaruint ^ doit satisfaire aussi & I'^uation <t>(ao\ <h\'"(hi'\ a?, -/iic + y) = <^(ao, Oi, ...a„; «, y); (F) et r^proquement, toute fonction ^ homog^ne par rapport aux coeffieients et aussi par rapport aux variables, qui satisfait k ces ^nations (X, Y), sera un covariant de la fonction donn^. Examinons d'abord T^uation (X) que je repr&ente par ^' = 0. Soit pour le moment, Oi'— ai = Xai, a,' — a, = \aa, &c., alors on aura, comme k Tordinaire, T^quation symbolique oh les quantity Oi, Os, &c., en tant qu'elles entrent dans cti, a,, &c., ne doivent pas 6tre affect^ par les symboles da^, 3a,, &c. de la differentiation. En substituant les valours de a^, a,, ...,et en ordonnant selon les puissances de \, cette ^nation donne ah les symboles D, Di, &c. sont donn^ par Q = ao3«, + 2aiaa, . . . + naj^ida^, et les quantity Oi, a,, &c., en tant qu'elles entrent dans les symboles Q, Di> &c, ne doivent pas Stre affect^ par les symboles da,, do,, &c. de la differentiation. II est assez remarquable que T^quation symbolique peut aussi 6tre ^rite sous la forme plus simple 0' = e^D-y^.) <^, 166 NOUVELLES RECHERCHES 8UR LE8 CO VARIANTS. [131 oh lea quantit^s Oi, a,, ..., en tant qu'elles entrent dans le symbole D, sont cens^es affect^s des symboles do,, d^,, &c. de la diff(^rentiation ; de mani^re que dans le d^veloppe- ment, D'.0 par exemple, signifie D . D0, et ainsi de suite. Je ne m'arrSte pas sur ce point, parce que pour ce que je vais d^montrer de plus important, il suffit de faire attention a la premiire puissance de X. D'ailleurs Tintelligibilit^ des ^nations dont il s'agit, sera facilitde en faisant les d^veloppements et en comparant les puissances correspondantes de \. Cela donne par exemple: n«=g«+2n„ n»=n»+3nni+6n„ &c. oil les symboles D^ D^ &c. k gauche de ces Equations d^notent la double, triple, &c. rdpdtition de reparation D, tandis qxx'k c6t^ droit des Equations, les quantity Oi, a,,... &c., en tant qu'elles entrent dans les symboles D, Di, &c. sOnt cens^ ne pas Stre affect^es des symboles da^, do,, &c. de la diff(^rentiation. Dans la suite, si le contraire n'est pas dit, je me servirai des expressions D', D", &c. pour d^noter les repetitions de rop^ration, et de m6me pour les combinaisons de deux ou de plusieurs symboles. Cela etant, T^quation 0' = c^(D-i'»J ^ = donne <^={i+x(n-ya,) + j^*2^n-ya,)»+...}0, oil (□ — y3«)'.0 (JQ le r^pfete) equivaut a (D ""y3«)'(g ~y^«)^> ®^ *"^ d® suite. H faut d'abord que le coeflScient de X s'^vanouisse, ce qui donne (D— y3jB)^ = 0; et cette condition ^tant satisfaite, les coefficients des puissances sup^rieures s'^vanomssent d'elles- mSmes ; c'est-&-dire, T^quation (X) sera satisfaite en supposant que ^ satisfait k I'^quation aux differences partielles (D — ydx) <f> = 0, EIn posant « □ = ctn3«^_j + 2a^i3a^ . . . + noida^, on fera un raisonnement analogue par rapport k Tdquation (F); et il sera ainsi demontre que <f> doit satisfaire aussi k r^quation k differences partielles (D — «9y)^=0; done enfin, on a le suivant TH]£oBi3iE. Tout covariant ^ de la fonction (ao, Oi, ... an){x, y)~, satisfait aux deux Equations k differences partielles (n-ya,)0=o, (n-^y)<^=o, (.i) oil « 131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 167 et r^proquement toute fonction, homog^ne par rapport aux coefficients et par rapport aox variables, qui satisfait k ces ^uations, est un covariant de la fonction donn^. Par exemple, Vinvariant ^^clc — I^ de la fonction cux^ + 2hxy + cy* satisfait aux Equations et le catfariant = (ac — 6*)«'+(a9 — 6c)a?y + (63 — c*)y* de la fonction aa:^+Sba^y-\-3cxi^+dy* satis&it aux Equations (aa^ + 2Me + 3cad-ya,)0 = O, (3d0e + 2ca6+W«-ajay)<^ = O. n est clair qu'en ne consid^rant que les fonctions qui restent les mSmes en prenant dans un ordre inverse les coefficients Oq, Oi, ... On et les variables x, y, respectivement, les covariants seront d^finis par Tune ou I'autre des Equations (A), et qu'il n'est plus n^cessaire de consid^rer les deux Equations. Cela posd, on trouve assez facilement les comariants par la m^thode des coefficients inddtermin^. Mais il y a & remarquer une circonstance de la plus grande importance dans cette throne, savoir, que Ton obtient de cette mani^ un nombre d'^uations plus grand qu'il n'en faut pour determiner les coefficients dont il s'agit Ces Equations cependant, ^tant li^es entre elles, se r^uisent au nombre n^cessaire d'^uations ind^pendantes. Cherchons par exemple pour la fonction a^ + 3ha^y + 3cd7y' + dy* un invariant ^ de la forme = ila«* + 5a6cd + Ooc' + Gh'd + D6«c», contenant les quatre coefficients ind^termin^s A, B, C, D, £n substituant dans T^quation (085+ 263c + 3cdd) ^ = 0, on obtient (3C + 25) a6»d + (3B + 6(7 + 2Z))a6c> + (6^ + 5) ac»d + (3(7 + 42)) 6»c = ; Of les quatre ^nations donn^es par cette condition, se rdduisent k trois Equations ind^pendantes, de sorte qu'en fsdsant par exemple -4=— 1, les autres coefficients seront d^tennmds, et Ton obtient le r&ultat connu : (^ = - d^« + 6a6cd - 4ac» - 46»d + Wc\ La circonstance mentionn^ ci-dessus s'oppose k r&oudre de la mani^re dont il ^^t, le probl^me de trouver le nombre des invariants d'un ordre donn^: probl^me ^ a toujours brav^ mes efforts. Avant d'entamer la solution des Equations {A)y je vais d^montrer quelques propri^t^s ^^rales des covariants, et des invariants. Pour abr^ger, je me servirai du mot pesanteur, en disant que les coefficients Oo, Oi, &c., ont respectivement les pesanteurs — ^n, 1 — ^n, ^) que les variables x, y ont respectivement les pesanteurs ^, — ^, et que la pesanteur 168 NOUVELLES RECHEECHES 8UR LES OOVARIANTa [l3l d'un produit est ^gale k la somme des pesanteurs des facteuis. Cela pos^, je dis que tout covariant est compost de termes dont chacun k la pesanteur z&o. Pour d^montrer cela^ j'^cris: (n-a0y)(n-ya,) = nn-ya,D-aj0yn+«ya.ayH-«a,; cela donne « or, en faisant attention aux valours de D, D, savoir gn = (DD) + nooaa^ + 2(n - l)a,a«^ ... + n 1 a^,a«_^, oil, en formant les produits (DU), (UD)) I^ quantit^s Oo, Oi, ... o^ sont cens^ non affect^s par les symboles da^,da^,... da^ de la difii^rentiation, on en tire nn-nn = nao3a +(n-2)oi9a ... — nonda = - 2 {(0 - hn)aoda^ + (1 - ^n)a,da^ . . . + (n - in)anda,} = - 26, en reprdsentant par 6 Texpression symbolique entre les crochets. De \k enfin on obtient : Or en supposant les deux parties de cette ^nation symbolique appliqu^ au covariant ^, la partie gauche de T^uation s'^vanouit en vertu des ^nations (A) et T^uation se r^uit k (8 + ia«x-iyay)0 = O; (B) ce qui est une nouvelle Equation k differences partielles, k laquelle satisSut le covariant <!>. II est ais^ de voir que cette Equation exprime le th^r^me dnoncd ci-dessus, savoir que tout covariant est compost de termes de la pesanteur ziro. n suit de Ik, en considdrant un covariant <^ = (ilo, Au ... At)(x, yy qu'un coefficient quelconque At aura la pesanteur i — ^8, ou bien que les pesanteurs ferment une progression arithm^tique aux diff(^rences 1, et dont les termes extremes sont — i«, +^s. Substituons maintenant cette valeur de <f) dans les ^nations (A). La premie ^nation donne d'abord: nilo = 0, DAi^Aoy DA^^iAu ...nA, = sA^^ (a) Cela est un syst^me qui ^quivaut aux deux Equations n'.il, = 0, = y'.e°f.il, (cT) 131] NOUVELLES RECHERCHES 8UR LES CO VARIANTS. 169 De mSme, la seconde ^uation donne BTSt&me qui ^uivaut aux deux Equations [!l'+Mo = 0, 4>^afe^».Ao (/SO On voit que A^ satisfait aux deux ^uations nA = 0, 6'+^ilo = 0, (7) et en supposant que cette quantity soit connue, on trouve les autres coefficients ^1, A^ ...,Ag par la seule differentiation, au moyen des ^uations 0^). Or cela dtant, je dis que les ^nations (a) seront satisfiedtes d'elles-mSmes. "En. effet: des Equations Dilo=0, [!lulo=«ili on tire 00^=0, DDA^^sDAu et de ]k (DD - nn)ilo = -«nili. Or nous avons d4jk vu que CD — 00 = 26, et TAjuation (B) donne B.Aq+^s.Ao^Oi done r^uation (DO — OiJ)ilo = — «0-4.i se r^uit k Ao=QAi: ^nation du systfeme (a). De la mSme mani^re on obtient les autres ^nations de ce syst^me. On pent dire que Ton aurait pu determiner ^galement le coefficient Ag au moyen des ^nations D4. = 0, D'.As = 0, (8) et de ]k les coefficients il^i, ... Aq par les ^nations (a). Prenons par exemple un covariant (Ao, Ai, A^) (a?, y)' de la fonction cubique (w* + 36a^ + 3ca?y* + dy*. A^ doit satisfaire aux deux ^nations (adt + 2hdc + ^d)Ao = 0, (36a« + 2cdt, + ddcfAo = 0. Ces Equations sont en effet satisfiedtes en mettant Ao = ac — b^. On a done les Equations 2Ai = {3bda + 2cdb + ddc)Ao, A^ = (Sbda + 2ddt + ddc)Au pour determiner Ai, A^; ce qui donne 2-4i = ad — 6c, il, = 6d— c", et on est conduit unsi au covariant mentionn^ ci-dessus, savoir k (ac - b^)a^ + (ad -be) ay + (bd - c^) y\ Soit maintenant ai^da^-'Of^^yda^ ... ±r3a.=A, on aura nA = (gA), AD=(AD)-y||. C. n. 22 170 NODTVELLES RBCHERCHES SUR LES COVARIANTS. [131 oil dans (DA), (AD) les quantity Oq, Oi, ... sont cens^ non affectdes par lea symboles da,> do,, &C. de la di£fdrentiatioD. Cela donne DA-AD = y^. Or 3aj A — Adx = ^ , done : (D-ya.)A = A(D-ya,), et de m6me: (n-aj0y)A = A(n-a»yX Appliquons ces deux Equations symboliques k un covariant 0. Les termes k droite s'^vanouissent k cause des ^nations (A), et Ton obtient les deux ^nations . (D-ya«)A0=O, (n-'/cdy)A4>^0, o'est-ji^dire : A0 sera aussi un covariant de la fonction donn^e. Par exemple de Vinva- riant on tire le covariant savoir : (- a^ + 3a6c - 2i» )«» -3( aM-2ac«+ 6«c)aj^ + 3( acd-26«dH- 6c«)ay« - (-a(?+3acd-2c» )y»; r&ultat ddjd. connu. Essay ons maintenant k int^grer les ^nations {A)\ savoir: (n-y9«)0-O, (D-aj0y)^ = O. Pour int^grer la premiere, je reviens k une notation dont je me suis ddjij servi dans ce mdmoire et j'^cris Oj' ssOi + XOo, Oa' = a, + 2\ai + X'Oo, On' = On + w\a»-i • • • + X**ao. 131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 171 En faisant X = , ce qui donne a^ = 0, on voit sans peine que Ton satisfera ^ r^uation, en mettant pour <^ une fonction quelconque de quantitds Oq^ a^, ... a^^ ^ + ^y> y \ 1® nombre de ces quantity ^tant n + 2. Et cela est la solution gdndrale de r^uation. Ce r&ultat doit 6tre substitute dans la seconde Equation, savoir dans (U — ^y)<^=0. Four cela, imaginons que les quantitds Oo, Oi, ... On, ^, y soient exprimdes en fonction de Oo'f CLtt ..• fltn'i ^> y et Ci ; puisque ^ est fonction des seules quantitds Oq', a^ ... On', X, y, r^uation r^sultante doit 6tre satisfiedte, quelle que soit la valeur de Oi. Or on trouve que cette Equation r&ultante a la forme L + Mai=:0: done il faut qu'on ait k la fois les deux ^nations Z = 0, M=0. (Je renvoie k une note les details de la r^uction.) En demi^re analyse, et en remettant dans les ^nations X — 0, M = les quantit^s Oo, a,, ..., On au lieu de Oo^ o^^ ..., an\ je trouve les r^sultats suivants tr^ simples, savoir, en ^crivant e = (0 - ^n)a,da^ + (2 - ^n)a,d^^ + (3 - in)a,a«^ . . . + (n - in)anda^ : •D = (n - 2)a^da + (n - S)a,da ... +and, «-i" Les ^nations dont il s'agit sont {(n-l)a,en-ya,)-aoCn-^,)}^=0, (C) (8 + ia«,-Jyay)^=0, (D) et il y a & remarquer qu'on obtient T^quation (C) en ^liminant entre les ^nations (A) le terme doi^; ^t puis, en mettant ai = 0, on tire I'^uation (D) de T^uation (B), en y mettant de mSme Oi = 0. II y a & remarquer aussi que la fonction <f> qui Kitis&it aux ^nations (C, D), est ce que devient un covariant quelconque <^, en y ntettant Oi^O. On obtient d'abord la valeur g^n^rale en changeant Oo, a,, ... , On en ^'> 0,', ..., On^ et en mettant apr^ pour ces quantit^s leurs valeurs en termes de ^f Oi, 0,, ..., On- La solution du probl^me des covariants serait done effectude si Ton pourrait int^grer les Ajuations (C, D). Or la quantity Oq entre dans Tdquation ((7) comme constante, et Ton voit sans peine que cette ^nation pourra 6tre int^grde en mettant Oo = 1 ; puis, en ^rivant dans le r&ultat — , — , ... — au lieu de Oj, 0^, .., a^ et en multipliant par une puissance qnelconque de Oq, le r&ultat ainsi obtenu, serait composd de termes de la mdme t^nteur; et en choisissant convenablement la puissance de Oo, on pourrait faire en sorte que ces termes fussent de la pesanteur z4to. Mais Tdquation (D) ne fait qu'exprimer que la fonction ^ est compos^e de termes de la peaanteur z^ro; le rdsultat obtenu de ^ mani^re dont il s'agit, satisfera done par lui-m6me k Tdquation (D), et il est pennis de ne fiedre attention qu'& I'^quation (C). Dans la pratique on intdgrera cette 22—2 172 N0UVELLE8 BECHEBCHES 8UR LES GOV ABI ANTS. [131 ^nation en ayant soin de faire en sorte que les solutions soient de la peswnJteur z^ro, ce qui peut Stre effectu^ en multipliant par une puissance convenablement choisie de Oo. Et puisqu'en faisant abstraction de cette quantity Oo, T^quation {(J) contient n + 1 quantit^s variables, savoir a,, a,, ...,an, x, y, la fonction ^ sera une fonction arbitraire de n quantity ; et en supposant que cette fonction ne contienne pas les variables w, y (cas auquel serait ce que deviendrait un invariant quelconque en y mettant Oi = 0), ^ sera une fonction arbitraire de n — 2 quantity. La mSme chose sera ^videmment vrai, si Ton r^tablit la valeur gdn^rale de a^ : done tout invariant sera une fonction d'un nombre n — 2 dUnvariantSy que Ton pourra prendre pour primitifs; et tout covariant sera une fonction de ces invariamts primiti& de la fonction donn^ (laquelle est dvidemment un de ses propres covariants), et d'un autre covariant que Ton peut prendre pour primitif. Cela ne prouve nullement (ce qui est n^nmoins vrai pour les invariants, k ce que je crois) que tout invariant est une fonction rationnelle et int^grale de n — 2 invariants convenablement choisis, et que tout covariant est une fonction rationnelle et int^grale (ce qui en effet n*est pas vrai) de ces invariants, de la fonction donn^e, et d'un covariant convenablement choisL Le cas n = 2 fait dans cette th^rie une exception. On sait qu'il existe dans ce cas un invaria/nt, savoir ac^l^ qui, selon la th^rie g^n^rale, ne doit pas exister, et il n'existe pas de covariant, hormis la fonction donn^e elle-mSme. Or cette particularity peut 6tre aisdment expliqu^. Le cas 71 = 3 rentre, comme cela doit 6tre, dans la th^rie g^n^rale. En effet, il existe dans ce cas un invariant, savoir la fonction — a'cP + 6ahcd + 4ac* — 46'd + 36'c* ci-dessus trouv^, et tout covariant de la fonction peut 6tre exprim^ par cet invariant de la fonction donn^ elle-mdme, et par le covariant (oc — 6*)«' + (ad — 6c)a;y + (M — c*)y' ci-dessus trouvd II en est ainsi par exemple pour le covariant de troisi^me ordre par rapport aux variables et aux coefficients; car en repr^ntant par ^ le co- variant dont il s'agit, par H le covariant du second ordre, par u la fonction donn^ aa? + Zbah/ + 3ca;y' + dy* et par V Vinvariant, on obtient I'^uation identique ^' + Dw* = — 4jEr". Je £aas mention de cette ^nation, parce que je crois qu'elle n'est pas g^n^ralement connue. Je vais donner maintenant quelques exemples des Equations (C et Z>). Soit d'abord 71=3, et supposons que 4> ^^ contienne pas les variables x, y: ^ sera une fonction de a, c, d, et les Equations reviendront k {Qd'dd - adde)^ = 0, (- 303^ + 09^ + 3(©d)^ = 0. Les quantit^s oc*, a*d^, dont chacune est de la pesanteur ziro, satisfont par 1& d. la seconde Equation, et en mettant ^ = ila'cP + Coc', on obtient 4il — C=0, en vertu de la premiere Equation; ou en faisant il = — 1, cela donne C= — 4; de Id. on tire ^ = — a*cP — 4ac*, et la solution gdn^rale est $ = jF(— a'cP — 4ac*), F ^tant une fonction quelconque. La formule plus g^n^rale ^^F{a, ^a^d^—^Kuf) satisferait sans doute k la 131] NOUVELLES RECHERCHES SUR LES CX) VARIANTS. 173 premie ^uation, mais pour que cette valeur satis&sse k la seconde ^uation, il faut que la quantity a, en tant qu'elle n'est pas 'contenue dans — a^c^ — isoc^, disparaisse. Ainsi la valeur donn^ ci-dessus, savoir ^ = jF(— a*(? — 4ac*), est la solution la plus gt^n^rale des deux ^nations. Ecrivons a, c , d 1 — r au lieu de a, c, d, et 6 au lieu de $, nous obtenons : 4> = F{'-a?d} + 6abcd - 4ac» - 46W + 36V); ce qui est Texpression la plus g^n^rale des invariants de la fonction aic*+3&ij^+3ca:^*+y', et Ton Yoit que tous ces invariants sont fonctions d'une seule quantity que nous avons prise ci-dessus pour Vinvariant de la fonction de troisi^me ordre dont il s'agit Soit encore n = 4, sera une fonction de a, c, (2, e qui satisfait aux Equations {2acBc + (a« - 9c*) da - licdde] ^ = 0, {-2aaa + d3d + 2eae}^=0, dont la solution gdn^rale est = F{ae + 3c*, axie — acP — c*), F ^tant une fonction quel- conque. On voit par Ul qu'il n'existe que les invariants ind^pendants ew — 4cd + 3c*, ace + 2icd — ocP — 6*6 — c*. Ce r&ultat est connu depuis longtemps. Soit enfin n = 5, <^ sera une fonction de a, c, d, e, f qui satisfait aux Equations {3adde + (2ae - 12c*) da + (a/- 16cd) de - 20cedf] ^ = 0, On sait qu'il y en a une solution de quatri^me ordre par rapport aux quantity a, c, d, e, f\ et en prenant la fonction la plus g^n^rale dont les termes ont la pesanteur z^ro, on aura: = ula*/» + Bacdf+ Cac^ + Bad^e + Ec^e + F(M^ : fonction qui satis&it d'elle-mSme k la seconde Equation. En substituant cette valeur dans la premiere ^nation, on trouvera que les coefficients A, B, &c. doivent satisfaire i ces sept ^nations : 2£ + 2C-40il = 0, 35 + 2) = 0, 3C + 42) = 0, -125 + iE' = 0, QiF- 24Z> + 4^*- 32C - 205 = 0, 6i^-162)=0, - 24i^- 16iE'= 0, <I^ se r^uisent cependcuit (ce que Ton n'aurait pas facilement devin^ par la forme des ^juations) k cinq Equations ind^pendantes. En faisant done il = 1, on trouve aia^meiit les autres coefficients B, G, &c. et on obtient ainsi : ^ = a*/» + 4acd/+ 16acc* - 12ad*e + 48c»c - 32c*d* : ▼aleur qui pent 6tre tirde d'une formule prdsent^e dans mon m^moire sur les h3rper- d^rminants, [161 ^^ y f^^dsant 6 = 0. 174 NOUVELLES BECHERCHES 8UR LES COVARIANTfiL [l31 J'ai donn^ cet exemple pour faire voir qu'il serait impossible de deduire du Dombre suppose connu des coefficients inddtermin^s qui correspondent, k un ordre donnd, le nombre des invariants de ce mSme ordre. II est done inutile de pousser plus loin cette discussion. Note 1 8ur Vintigration des iqwiUons {A). En ^rivant comme ci-dessus: n = afia, + Soiaao. ... + nan-i3a,, n = noxda. + (n - \)a^da, ... + CLn\_^^ il s'agit de trouver une quantity ^, fonction de Oq, Oi, ... an, a? et y qui satisfasse k la fois aux ^nations (n-a?ay)0=o. Pour int<^grer ces ^nations, j'^ris, comme plus haut : do' = 00, a/ = ai + Xoo, ai^^a^-^ 2Xai + Va©, On' = On + nXa,i_i ... +X*ao, et aussi of = x — \y, y' = y. Cela pos^, je fais remarquer d'abord que ^r- = ao'» ^=2ai', et ainsi de suite. Eln consid^rant X comme fonction quelconque de Oo, Oi, ... On, et en supposant que ^ soit une fonction de Oo', Oi^ ... a^, oiy y\ on parvient assez facilement k r^quation identique (D — y9aj)^ = (l + D^) (III'~y'3«')0> ^^ Q' ©st ce que devient D, en y ^rivant Oo', Oi', ... a^ au lieu de Oq, Oi, ... On. Nous pouvons done satisfaire k la premiere ^nation, en determinant \ au moyen de l + nx=0: &}uation qui serait satisfaite en ^rivant \= — — , ou, si Ton veut, en determinant X par a^^O. Done, en supposant toujours que X ait cette valeur, 4> sera une fonction quelconque de Oo', a,', ... a^, of, y\ c'est-i-dire d'un nombre n + 2 de quantit^s. Ce sera done \k (comme on aurait pu facilement prdvoir), la solution g^n^rale de la premifere Equation. Or en consid^rant <f> comme fonction de Oo', a,', ... a„\ of, yfy ou, si Ton veut, de Oo', a^, a,', ... a^, of, y' (oil Oi' = ai + Xo© = 0), et en 131] NOUVELLES RECHERCHES SUB LES COVAMANTS. 175 sabstituaDLt cette valeur dans T^quation (D — a^y)0 = 0, on voit d'abord que la variation de la quantity X fonmit au r^sultat le terme („a.gH«-l)«.£)(n'-j^.)*; et puisque na ^ + (^""^)^^'^ r^uit & n-^-(n — 1) — ^, ou enfin & V-^^ —, ce terme devient Le terme —oi'dy,^ se r^duit & — (a?' + XyO(""^«'+3y')^» savoir & et en mettant pour un moment M— woi(9fl,' + X9aj' +X*3|,^/) + (n-l)a2( ao,' +nX«-^aa;) + an( 9a^/+wXa«^,), nous obtenons c'est-i-dire €•0 Or en supposant que D' est ce que devient D en y ^crivant a©, Oi', ... a»' au lieu de Oo* ai, ... On, et en posant e' = (0 - in)ao'aa.' + (1 - in) Oi'ao,^ + . . . (n - iii)an'aa;, on obtient, aprte avoir fait une r^uction un pen pdnible : M^ + X«n'<^ = n'0 + 2X8'^, (en effet les coefficients de da,'<^, 'da{^ &c. aux deux cdtes de cette ^nation deviennent les mSmes aprte des r^uctions convenablea) Done enfin on a (D - a0v)* = ([i' - a;^,^)^ - ^^^^^^ (D' - y'3^)^ + 2X(e' + J<c^^ - iy'8^)^ = 0. ott bien, puisque cette Equation doit Stre satis&ite inddpendamment de la quantity X (qui fleule contient Oi), elle se decompose dans les deux ^nations {aii^ - x'Z^) - (n - 1) ai{U' - j/a^)} <^ = 0, 176 NOUVELLES RECHERCHES SUB LES COVARIANTS. [131 lesquelles, en y mettant d'abord a^^O, puis en remettant Oo, a,, ..., On, a?, y an lieu de Oo', cLtf ... , Oni «^» y'l et en ^crivant 0, O, •Q, d au lieu de ^, O, D, D, donuent en effet les Equations (C, D) dont je me suis servi dans le texte. Note 2. Je vais r^sumer dans cette note quelques formules qui feront voir la liaison qui existe entre les invariaiUs d'une fonction de fi-i^me oidre et de la fonction de (n — l)i&me ordre que Ton obtient en r^uisant & z^ro le coefficient de y*^, et en supprimant le facteur x. IV convient pour cela de consid^rer une fonction telle que (Oo, Oi, ... an){^,y)n = a^ + (h^~^y ... +a„y~, dans laquelle n'entrent plus les coefficients numdriques du bindme (1 + x)\ Ecrivons {a^,Q^, ... a„)(a?, y)« = ao(a?-aiy)(a?-a^) ... (a?-any); je t&che d'abord & repr&enter les invariants au moyen des racines ai, a„...,a„, et j'^tends pour le moment le terme invariant k toute fonction, sym^trique ou non, des racines qui ait la propri^t^ caract^ristique des invariants: fonctions qui jusqu'ici ont 4t6 consid^r^es tacitement comme rationnelles par rapport aux coefficienta Mettons d'abord V = Oo^^ioi - a^fioi -«,)»... ((V-i - OnY ; cette quantity V qui, ^gal^ k ziro, exprime T^galit^ de deux racines, et que je vais d^rmais nommer le Discriminant de la fonction, sera une fonction rationnelle des coefficients, et d'un invariant proprement dit. Mais de plus, toute fonction telle que (Oi — a,)**(ai — a,)*, ..., dans laquelle la somme des indices des facteurs qui contiennent ttx, celle des indices des facteurs qui contiennent a,, &c. sent ^gales, sera un invariant; et en r^unissant ces fonctions, pour trouver une somme en fonction 8}anetrique des racines, on obtiendra des invariants proprement dits. Cela soit dit en passant. Pour le moment il suffit de prendre les invariants les plus simples, savoir ceux de la forme («! - tt») (g| - q^) (ax - a,) (a, - fl^i) ' lesquels en effet sent des rapports anharmoniques de quatre racines, prises k volontd. Soient Oi» Os, ••.>0»-i 1* fonction qui vient d'etre ^crite et les fonctions que Ton en tire en mettant a^, fl^j, ...,an au lieu de a^. Les fonctions V, Q^, Qj, ...,Q,i_, seront des invariants ind^pendants, et le nombre de ces invariants est n — 2. Done, tout autre 131] NOUVELLES RECHERCHES SUB LES CO VARIANTS. 177 invariant sera une fonction des quantity V, Q^, Q,, ..., Qn^, Soit maintenant an = 0» et cEn 1ft racine qui devient dgale k z4ro. Lea qucuitit^ Qi, Qa* •••> Qn^ seront toujours des rapports anharmoniques de quatre racines de T^uation da (n — l)i6me ordre. II n'y aura que la seule quantity Qn^ qui change de forme, et elle ne sera pas un intfariant de la fonction du (n — l)i^me ordre. On voit aussi d'abord que le discriminant V se r^uit k a*n-iVo» en exprimant par V© le discriminant de la fonction du (n — l)ifeme ordre. (Cest je crois M. Joachimstbal qui a le premier remarqud cette circonstance.) Done, en supposant an = 0, Vinvariant de la fonction du n-i^me ordre deviendra une fonction de a^n-^o, Qi, Q,, ... Qn^ et d'une quantity X qui n'est pas un invariant de la fonction du (n — l)ifeme ordre, mais qui sera toujours la mSme quel que soit Tinvariant dont il s'agit. En consid^rant les invariants proprement dits de la fonction du (n — l)i^me ordre, on pent former avec ces irivariants des quotients /i, /„ ...,/,^-^ du degr^ z^ro par rapport aux coefficients. Nous pouvons remplacer par ces quotients les quantit^s Qii Qt) "-iQnr-ii et dire que Vinvariant de la fonction du n-i^me ordre, en mettant an = 0, deviendra une fonction des quantit^s a'»_iVo, A, /,, ..., /,^-^ et X. Ces thdor&mes auront, je crois, quelque utility pour les recherches ult^rieures: je les laisse k c6t^ maintenant, et veux presenter une m^thode assez simple pour calculer les discriminants. Four cela je remarque que les Equations {A\ en changeant, comme nous venous de le £Edre, les valours des coefficients, donnent pour les invariants : (naoda, + (n - l)aida, . . . + a^^ida) = 0, (aida. + 2a,'aa, • • • + nanda^)4> = ; et ces ^nations seront satisfiedtes en mettant pour <f> le discrimina/tU V. Or, pour fln=0, la fonction V devient a*«_iVo, ou, si Ton veut, — a'n-iV©; done V sera g^nerale- nient de la forme oil On*^^ est la puissance la plus dlev^ de an- Done, en supposant que Vq soit connu, et en mettant la premiere des Equations Writes ci-dessus sous la forme (^+a»_i3a^)V=0, 0^ l'=na©9ai + (n — l)oi3fl, ... +2aw»,3a , on obtiendra par la seule diff(^rentiation les efficients B, C, &c. EIn effet, cette Equation donne a^iB^F{a*^,V,l %i^,G — F{B\ San^,D = ^ F(C) ; et ainsi de suite. En supposant par exemple n = 3, consid^rons la fonction du troisi^me ordre ^^ ixfmminani de okc* + ^xy + yy* sera 4ay — /S*. Nous avons alors an. 23 178 NOUVELLES BECHERGHES SUB LES CJOVABIANTS. [131 et en mettant ^=3a9^ + 2i98y, B, C seront donn^ par c'est-Ji-dire B ^ ISafiy - *fi*, C = -27a», et de ]k: V = - 27 fl?S« + ISa/SyB - 407' - 4/9»S + ^Sy : valeur qui correspond en efifet k la forme ordinaire V = -a«* + 6a6od-4ac«-4W + 86V, en changeant d'une mani^re convenable les coefficient& Londres, SUme Buildings, 23 Fhr. 1852. 132] 179 132. REPONSE A UNE QUESTION PROPOSEE PAR M. STEINER (Au%abe 4, Crelle t. xxxi. (1846) p. 90). [From the Jcumal far die reins und angewandte Mathematik (Crelle), torn. L. (1855), pp. 277—278.] En partant des deux th^rfemes : I. Qu'U existe au moins une surface du second ordre qui touche neuf plans donn& quelconques ; n. Que le lieu d'intersection de trois plans rectangles qui touchent une surfisM^e ^^ second ordre est une sphere concentrique avec la surface, tandis que pour le para- ^loide cette sphere se r^uit It un plan, ^ Steiner suppose le cas d'un parall^epipkle rectangle, ou mdme d'un cube P ^^ d'un point quelconque D, par lequel passent trois plans rectangles. Les six plans ^^ parall^lepipMe P et les trois plans qui passent par le point D seront touchy d'une ^^'fece F du second ordre (I.), et les huit angles E du parall^epipfede P et le point ^ doivent done se trouver tons les neuf sur la surfiek^e d'une sphere, ou dans un P^^^ (XL). Les huit angles E sont en efifet situ6s sur la surfiek^e d'une sphere, ^^t^rmin^ par eux ; mais le point D ^tant arbitraire, ce point en g^n^ral ne sera P^B situ^ sur cette surface sph^rique, de mani^re que les neuf points 8E et D ne ^<t>nt ntuds, ni dans une surface sph^rique, ni dans un plan; ce qui ne s'accorde P^ avec le th^orfeme II. Cela etant, M. Steiner dit, qu'il y a It prouver que la ^xitradiction n'est qu'apparente, et que tout cela n'affaiblit pas la validity g^n^rale des deux th^rfemea II s'agit de savoir ce que devient dans le cas suppose par M. Steiner la surface du second ordre qui touche les six plans du parall61epipMe P et les trois plans qui 23—2 180 lUiPONSE 1 UNB QUBSnON PROPOSJ^ PAR M. 8TEINER. [132 passent par le point D. Cette surfSeu^ sera en eifet la conique aelon laquelle Vinfini, consicUrS comme plan, est coupi par un cdne ditermind, pris la position du sommeL En efifet, menons par un point queloonque de Tespace trois plans parall^les auz plans du parall^epipMe P, et par le point D trois autres plans parall^les k ces plana Ces six plans seront touch& (en vertu d'un th^rfeme oonnu) par un cdne ddtermin^ du second ordre, et on pent dire que ce cdne, quelle que soit la position de son sommet, rencontre Tinfini, consid^rd comme plan, dans une seule et m£me conique (cela n'est en effet autre chose que de dire que deux droites parallfeles rencontrent I'infini, con- siddr^ comme plan, dans un seul et m£me point). Le cdne dont U s'agit aura la propridt^ d'dtre touchy par une infinite de syst^mes de trois plans rectanglea En effet: le plan passant par le sommet, et perpendiculaire k la droite d'intersection de deux plans tangents quelconques sera un plan tangent du cdne; les plans d'un tel syst^me seront aussi des plans tangents de la conique mentionn^e ci-dessus: done le sommet du cdne sera le point d'intersection de trois plans rectangles de la conique; et ce sommet ^tant un point entiferement ind^termin^, le lieu de Tintersection des trois plans tangents rectangles de la conique, sera de mdme absolument inddtermin^, ou si Ton veut, ce lieu sera Tespace entier pr&s les points k une distance infinie. La contra- diction apparente dont M. Steiner parle, a par cons^uent son origine dans Tind^ter- mination qui a lieu dans le cas dont il s'agit. Dans tout autre cas, le point d'intersection des trois plans rectangles de la surface du second ordre est parfidtement ddtermin^, et les th^rfemes I. et IL sont tons deux l^gitime& 133] 181 133. SUE UN THEORilME DE M. SCHLAFLL [From the Journal fQ/r die reine wnd angeivandte MathemaiUe (Crelle), torn. L. (1855), pp. 278—282.] On lit dans (§13) d'un m^moire tr^ int^ressant de M. Schlafli intituld "Uber die Resultante eines Systems mehrerer algebraischer Oleichungen" {M4m. de VAcad, de Vienne, t TV. [1852]) un tr^ beau th^r^me sur les R^sidtants. Pour £Eure voir plus clairement en quoi consiste ce th^reme, je prends un cas particulier. Soit F=ar» + 2)ftry + 7y" =(«, A y)(x, yy. Je fiEds p=a?, ? = «yi ^=y'i et je forme les op^rateurs 81= faa + M + iCSc, lesquels, operant sur U, donnent L'op^teur operant sur V, donne 182 SUB UN THltoBilME DE M. SCHLAFU. [133 Cela ^tant, soit ^==0 le rSstdUmt des ^uations U^O, F=0, c'est-iirdire I'^uation que Ton obtient en ^liminant w, y entre les ^uations Cr«0» F=0, ou autrement dit, soit ^ le resultant des fonctions U, V. Pour fixer les id^ j'^cris la valeur de ce r&idtant comme suit: ^s a, Sb, Sc, d a, db, 3c, d «, 2)8, 7 «, 2/8, 7 Je suppose que les op^rateurs 9(, S, € op^nt sur le resultant ^, ce qui donne les fonctions a*, «*. S0, ou en ^rivant pour 81, S9, S leurs valeurs : et en consid^rant ces expressions comme des fonctions de {, 17, (T, j'en forme le resultant ^, savoir Ce resultant O contiendra le carr^ de ^ comr/ie /(icteur; c'est ce qui donne, dans le cas particulier dont il s'agit, le th^or^me de M. SchlaflL Q^n^ralement, en supposant que Ton ait autant de fonctions U, V, TT, ... que d'ind^termin^s x, y, ^, .••» on pent supposer que p, q, ... soient des mon6mes aftf^tf^,... du mSme degr^ X (il n'est pas n^cessaire d'avoir la s^rie entifere de ces mondmes), et on pent former des op^rateurs 9(, 93, &c. en mdme nombre que celui des mondmes p, f, ... avec les ind^termin^es {, 17, ... , tels que ces op^rateurs SI, S, ..., operant sur les fonctions 17, F, IT, ... (chacun sur la fonction It laquelle il appartient), donnent ^(pf + 917 ...)**, ^(/>f + 917 ...)'*', &c.; t, ify &c. ^tant des mondmes de la forme x^tf^sf^ Cela ^tant, soit <f> le r&ultant des fonctions U, V, IT,...; en operant sur ce resultant ^ avec les op^rateurs 9(, 9,... et en formant ainsi les fonctions S(^, S^, ..., soit ^ le r&ultant de ces expressions consid^*^ comme des fonctions de {, 17, &c. ^ contiendra une puissance de ^ comme facteur, et en supposant que /i ne soit plus petit qu'aucun autre des indices /*, /*',...; 7r = /A/i'...; et o-= - + —> + ..., I'indice de cette puissance sera au moins a . Voil^ le th^rfeme g^n^ral de M. SchUlflL 138] 8UR UN THiOR^ME DE M. SCHLAFLI. 188 La demonstration donn^e dans le m^moire cit^ est, on ne pent plu8» simple et Aigsjite. E!Ue repose d'abord sur un th^r^me connu (d&nontr^ au reste § 6) qui pout dtre ^nonc^ ainsi; savoir, en supposant que les Equations {7=0, F&=0, ... soient satisfisdtes, on aura (prte un facteur ind^pendant de {, 17,...) ^ ^<l>^t(p^ + qfj...y, 8^ = ^ (pf + 517. ..>*', &C. Puis, elle est fond^ sur le th^rfeme d^montr^ (§ 12), savoir: le r&ultant des fonctions {oh /, /*,... sont des poljnidmes de degrds /*, /*',... en f, 17, &c., et p, g, ..., t, ^, ... des constantes quelconques) sera, en supposant que /i ne soit plus petit qu'aucun autre des indices fi, M^•••> et en posant 7r=/i/i'..., tout au plus du degr^ - par rapport aux quantity t, If, &a Voici cette demonstration, qui suppose aussi que le r&ultant ^ soit indecomposable, Supposons que les coefficients de U, F, IT, ... soient assujettis k la seule condition d'etre tels que le r&ultant ^ soit un infiniment petit du premier ordre, il sera permis de supposer que tous ces coefficients des ind^termin^ x, y, ... ne different des valeurs qui satisfont aux ^nations U^O, F=0, TT^O, ... que par des increments infiniment petits du premier ordre; le resultant if> sera un infiniment petit da premier ordre, mais toute autre fonction des coefficients, k moins qu'elle ne contienne nne puissance de ^ comme fiekcteur, aura une valeur finie, et toute fonction des coefficients infiniment petite de I'ordre k contiendra ^ comme facteur. Dans cette supposition les Equations S[^ = 0, S^ » 0, &c. deviendront : oil/ /',..> sont des polyndmes de degr^s /li, fi\ ... dont les coefficients sont des infiniment petits du premier ordre. En supposant toujours que /* ne soit plus petit qu'aacun autre des indices /i, /i^... et en posant 7r = fifi\.., o- — — + ->...> le resultant ^ fAf fit du syst^me sera tout au plus du degrd — par rapport aux quantit^s finies t, t^ .... Le d^ par rapport k tous les coefficients est o-; le degr^ par rapport aux coefficients <le /,/',... sera done au moins o- ; c'est-Jrdire, ce resultant sera un infiniment petit de I'ordre o- , ou enfin, ^ contiendra ^^"^'f^ comme facteur. Or les coefficients ^ U, F, Wf... (assujettis k la seule condition ci-dessus mentionn^e) etant d'ailleurs ^Mtraires, on voit sans peine qu'il est permis de &ire abstraction de la condition, et que <t contiendra en gdn^ral cette mdme puissance ^^^-^'i^ comme facteur; ce qu'il ^agisBait de d^montrer. 184 SUB UN THltoBilME DE M. SCHLAFLL [l33 Rien n'emp^he que ^ ne contienne une plus haute puissance que ^^^^^-i^ comme facteur, ou que ^ ne s'^vanouisse identiquement. On pent mdme assigner de plus pr^ que Fa (ait M. Schlafli, des cas o{i ^ s'^vanouit identiquement Soient m, m\ m'\ . . . les degr& de U, V, W,.,. par rapport i, a?, y, 2r,... , p = mmW ...,« = — + —,+ ^, ... , 771 771 771 if> sera du degr^ ^ par rapport aux coefficients de U. Soient aussi fi^, fi,,, ... les degr^s de celles des fonctions S(^, 9^, . . . , pour lesquelles les op^rateurs S[, S, . . . contiennent des diff^rentielles par rapport aux coefficients de U, p = — I — ... : pour ces fonctions les coefficients seront du degr6 — — 1 par rapport aux coefficients de U ; pour les autres ils seront du depr^ — . 4> sera done du degr^ (~ — l)p + ^(o- — p), = — o- — p, m \fn J m m par rapport aux coefficients d^ U, et ^-r^'^-'* sera du degr^ —a — p"^ (a j, c'est- k-diie du degrd *- . p par rapport aux coefficients de U, De m^me, en supposant que les lettres m', p', ... aient rapport It V, &c., O-s-^'^''* sera du degrd —,. p\ &c. lit Uf par rapport aux coefficients de F, &c. Si Tun quelconque des nombres — . — p, ^, . p', &c. est nigatif, et It plus forte raison, si leur somme 8 . — o- est negative^ 771 Uf Ut ^ doit s'^vanouir identiquement. En particulier, en supposant que le nombre des fonctions S[^, S^, ... (c'est-lt-dire le nombre des ind^termin^ ^, 17, ...) soit v^ on aura a->v- , et par cette ndson ^ s'dvanouira identiquement si - (o-— i/) est n^gatif, c'est- It-dire si v>a. Je ne parlerai pas ici des cas examine par M. Schlaffi, oil 4> con- tient comme facteur une plus haute puissance que ff^r^'i^. 134] 185 134 KEMAEQUES SUE LA NOTATION DES FONCTIONS ALGEBEIQUES. [From the Journal fWr die reine und angewandte McUhematik (CSrelle), torn. L. (1855), pp. 282—285.] Je me sers de la notation «', fi", y\ of\ P\ i\ pour repr^nter ce que j'appelle une matrice; savoir un systime de quantit6s rangdes en forme de carrS, mais d'ailleurs tout It fait indSpendantes (je ne parle pas ici des matrices rectangulairea). Cette notation me parait tr^s commode pour la th^rie des ^luations UnAiirea; j'^ris par exemple (f, V» ?, ...) = ( « * fi 9 7 > ••• /»" iO" -/' X«^> y» «> •••) pour repr^nter le syst^me des ^nations f = « x-^-fi y-^ry z ... , c. n. 24 186 REMABQUES SUB LA NOTATION DE8 FONCTIONS ALG^BRIQUES. [134 On obtient par Ik T^quation: (a?, y, z, ...) = ( a, /3 , 7 , ... r^Xfi Vf (T, a', 13'. y\ ... Q^'i ^', y, ... ••/» qui repr^nte le syst^me d'^uations qui donne x, y, z, ... en termes de {,17, (r> *.• > et on se trouve ainsi conduit k la notation — 1 « I ^ > 7 > ••• *'f ^'» 7'» ••• -,// jcy^ ^// de la matrice tnt^er«6. Les termes de cette matrice sont des fractions, ayant pour d^nominateur commun le determinant formd avec les termes de la matrice originale; les numiriiteurs sont les determinants mineurs form6s avec les termes de cette mSme matrice en supprimant Tune quelconque des lignes et Tune quelconque des colonnes. Soit encore (a?, y, z, ...) = ( tty Oy Cy... _/ r/ -/ tVy " t Cy... _// r// -// Cvy ^> Cy... Xa?, y, z, ...), on pent ^rire: .) = ( a', ^ , 7 1 ••• /s*. y.... /9". 7".... a', a", Of C f • .. Of C y . . . Of V y • . • A^f Jft Zf ...}, et Ton parvient ainsi k Yid6e d'une matrice compoaie, par ex. a , )8 , 7 , ... «'» ^'> y* ••• /»'' iO'' «,/' « > P , 7 > ••• Cvy ^> Cy... Cvy ^1 C f • • . d'f Vf (j'f ... On voit d'abord que la valeur de cette matrice compost est («» fi> 7> •••X^» ^'» ^"* •••)» (*» ^» 7> •••X^> 6'i 6''* •••)> ••• 134] BEICABQUES SUB LA NOTATION DBS FONCTIONS ALGJ^BIQUES. 187 oil (a, fi, 7, ...Xa, o,\ a"i ...) = «« + i8a' + 7a'' + ... . II faut fiaire attention, dans la com- position des matrices, de combiner les lignes de la matrice It gauche avec les cclonnes de la matrice It droite, pour former les lignes de la matrice compost. II y aurait bien des choses It dire sur cette th^orie de matrices, laquelle doit, il me semble, pr^c^er la th^rie de Diterminants, Une notation semblable pent dtre employee dans la th^rie des fonctions quad- ratiques. En effet, on pent d^noter par ( a , fi , 7 , ... *'> ^» Vf ••• M^' /y ^." Xf. % fXa?, y, ^) la fonction linwhlvniaire et de lib par la fonction qwxdraiiqaj^ ( a, A, ^r, . A, 6, /, . 5^, /, c,.. X«> y. -8:, ...)» oa^ + 6y* + c-?* + 2/y2r + 25r-^a? + 2Aa;y ... que je repr^nte aussi par (a, 6, c, .../, g, h, ...X^, y, -^f ...)". Je remarque qu'en g^n^ral je repr^sente une fonction rationnelle et int^grale, homog^ne et des degris m, m\ &a, par rapport aux ind^termin^es x, y, &c., a/, y', &c., de la mani^re suivante: (0X^» y,'"T{^y y', ...)"•' .... Une fonction rationnelle et int^grale, homog^ne et du degr^ m par rapport aux deux ind^termin^ d?, y sera done repr^nt^e par ( X^, yT' 24—2 188 BEMABQUES SUB LA NOTATION DES FONCTIONS ALQ]£!BBIQUES. [l34 En intFoduisant dans cette notation les coefficients, j'^ris par exemple (a, 6, c, d$x, yy, pour repr^nter la fonction tandis que je me sers de la notation (a, 6, c, cCJlx, yy, pour repr^nter la fonction (M5* + ba^ + cay* + dy*, et de mSme pour les fonctions d'un degr^ quelconqua tTai trouvd cette distinction tr^ commode. [In the foregoing Paper as here printed, except in the expression in the second line of this page, )( is nsed instead of y{ : it appears by a remark {CreUe, t. u, errata) that the mannsoript had the inter- laced parentheses J^ . Moreover in the manuscript ( ) was osed for a BCatrix, which was thus distingnished from a Determinant, but in the absence of any real ambiguity, no alteration has been made in this respect. In the reprint of subsequent papers from Grelle, the arrowhead }( ^^^ 3[ ^ ^^'^ instead of {J) . ] 135] 189 135. NOTE SUR LES C0VARIANT8 D'UNE FONCTION QUADRATIQUE, CUBIQUE, OU BIQUADRATIQUE A DEUX INDETERMINEES. [From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855), pp. 285—287.] La th^rie d'une fonction k deux ind^termin^es d'un degr6 quelconque, par example ( X^, yn depend du syst^me des cavaricmta de la fonction, lequel est cens^ contenir la fonction elle-mSme. Pour une fonction quadratiqiie le syst^me de covariants est (a, 6, cXa?, yA ac — 6*. Pour la fonction cubiqtie, le systfeme est (a, 6, c, dX^f, y)», (ac — h^, ad — bc, hd — (^){x, y)", (- a« + 3a6c - 26", -aM+2ac*-6^, ood-26« + 6c», ad>-3&cd + 2c»)(a?, y)\ - a*cP + 6a6cd - 4ac« - 46»d + 36»(J«, lonctions lesquelles, en supposant qu'on les reprdsente par U, J7, ^, Q, satisfont iden- ^^ement It I'&iuation 190 NOTE SUB LES COVAMANTS DUNE FONCTION QUADRATIQUE, [135 Pour la fonction biqiuidratique, le syst^me est (a, 6, c, d, cX«?, y)*, (ac-b', 2ad-2bc, ae + 2bd-3(f, 266-2cd, ce-d?Xx, y)*, ace + 26cd — ocP — l^e — c*, /^ - a»d + Sabc - 26», -a»c -2a6d +900" -66*c, -5a66 +15acd-106^, + lOa'd - 106>6, I (a?, y)», + 6ad6 + 106(? - 155ce, + a6» +26<fo -9c»c +6c(P, ^ +6e* -Sccfe +2d», et ces fonctions, en supposant qu'on les repr^nte par U, /, J7, /, O, satdsfont iden- tiquement It I'^quation J'ajoute k ce systfeme la fonction /» - 27/» = aV - 1 2a»6(fo» + 54a6*c6* - 6a6»d«6 -54ac»d« -276V + 366»C\P, ISaVc* +54a%eP6 -27a^ 180a6c'c26 + lOSabcd^ + 81ac% 646»d« + lOSb'cde - 646Vc qui est le discrimincmt de la fonction biquadratique. Pour donner une application de ces formules, soit proposd de r&oudre une ^nation quadratique, cubique ou biquadratique, ou autrement dit : de trouver un facteur liniaire de la fonction quadratique, cubique, ou biquadratique. U est assez singulier que pour la fonction quadratique la solution est en quelque sorte plus compliqu^e que pour les deux autrea En effet, il n'existe pas de solution sym^trique, It moins qu'on n'introduise des quantit^s arbitraires et superflues; savoir, on trouve pour facteur lin^aire de (a, 6, cX^, y)* Texpression (a, 6, cXa, iSXa?, y) + V- D .(i8«-ay), oil (a, 6, cX«, /8Xa?, y) denote aaa + b (ay + fix) + c/Sy. Pour la fonction cubiqrie, T^quation O* + D iT'* = — 4tH* fisdt voir que les deux fonc- tions 4> + I7V — D , ^ — ^V — D soiit Tune et Tautre des cubes parfedts. L'expression V {H^ + u-/ - o)} - y [H^ - u^ - m 135] CUBIQUE, OU BIQUADRATIQUE A DEUX IND^TERMHrflES. 191 sera done une fonction lin^aire de x, y; et puisque cette fonction s'dvanouit pour U=0, elle ne sera autre chose que Tun des facteurs lin&dres de (a, b, c, d)(x, yy. Pour la fonction biquadrabiqtie, en partant de T^quation j'^cris et je mets T^uation sous la forme (1, 0,-Jf, M)(IH, JUy=^-iP^. Done, en supposant que tJi, vr^, «j, soient les racines de T^quation (1, 0, - M, MXm, If = 0, OU plus simplement de T^uation t!r»-Jlf(t!r-l) = 0, ces expressions IH — vtiJU, IH — vrJU^ IH — vr^JU seront toutes trois des earr& de fonctions quadratiquea L'expression sera done une fonction quadratique, et on voit sans peine qu'elle sera le carr^ d'une fonction liniaire. Or cette expression s'^vanouit pour {7=0; done ce sera pr^eis^ment le earrd de I'un quelconque des facteurs lin^aires de (a, 6, c, d, e){x, yf. L'^uation identique pour les covariants d'une fonction biquadratique donne lieu aussi (remarque que je dois k M. Hermite) It une transformation tr^s ^l^gante de tinUgrdU dliptique |da?-^V(a, 6, c, d, e^x, 1)*. 192 [136 136. SUE LA TKANSFOKMATION D'UNE FONCTION QUADRATIQUE EN ELLE-M^ME PAR DE8 SUBSTITUTIONS UNfiAIRES. [From the Journal fWr die reine und angewandte Mathemoitik (Crelle), torn. L. (1855), pp. 288—299.] Il s'agit de trouver les transformations lin^aires d'une fonction quadratique ( )(«! y, ^» •••)* ^ eUe-mSme, c'est-lt-dire de trouver pour («, y, z, ...) des fonctions lin^aires de x, y, z, ... telles que En repr^ntant la fonction quadratique par (0X^» y» ^^ •••)* = o. A. 9* .•• h. b, J 9 ••• 9. • • • /. c , • • • Xa?, y, -er, ...)«, la solution qu'a donn^e M. Hermite de ce probl^me peut dtre rdsum^ dans la seule &|uation (x, y, z,...) = ( a, h,g, ... K b,f, ... fft Jf ^f ••• — 1 a, h-p, g + fi, ... h + v, 6, /— ^ ••• o, h + v, g — fi, ... h — p, b, f+\ ... — 1 a, h,g, ... A, 6,/, ... gf j$ ^f • • • )\*^ty»^$»")» oil X, /i, I', ... sont des quantit^s quelconques. 136] SUB LA TRANSFOBMATION d'uNE FONCTION QUADRATIQUE &C. 193 En effet, pour ddmontrer que cela est une solution, on n'a qu'^ reproduire dans un ordre inverse le proc^d de M. Hermite. En introduisant les quantity auxiliaires (i> V» K* •••)> ^^ P^^^ remplacer T^uation par les deux Equations ( ( a, A, g, ... A, 6, /, ... 9* ji c, ... a, A, ^r, . . . K ft, /, ... 5^1 ji c, ... Xa?, y, z, ...) = ( A^> y» ^> ••./ — \ a, A + i/, g — fi,... A — v, 6, y + x, ... * a. A -I/, ^r + A*, ... A + ^'i &> / — X, . . . Xf» ^» ?» •••) Xf» ^» ?» •••) qui donnent tout de suite d'abord et puis a: + x = 2f, y + y = 2i;, ^ + z = 2j; &c. On obtient par Ul: (OXx, y, z, ...)» = ( 0X2f-x, 2i;-y, 2?- z, ...)», =4(oxf. ^. ?, ...)»-4(oxf, ^> r....x^, y. -^^ ...) c'est-il-dire T^uation ( Xx> y. z, ...)»=( X^'» y. -^^ •••)"» qu'il s'agissait de verifier. Je remarque que la transformation est toujours propre. En effet, le determinant de transformation est a, h, g ... A, b, f ... 9,f, c ... —I h+v, b, y— X ... g-fi,f+\ c ... Oi h + v, g — fi ... h — v, b, y + X ... fl^ + A*, /-X, c ... — 1 a, A, ^r ... A, 6, / ... g> f, ... Or les determinants qui entrent dans les deux termes moyens, ne contiennent Tun ou Tautre que les puissances paires de X, fi, v, ... . Done ces deux determinants sont ^ux, et les quatre termes du produit sont rdciproques deux k deux; le determinant de transformation est done + 1, et la transformation est propra Pour obtenir une transformation improprey il faut consid^rer une fonction quadratique qui contient outre les indeterminees x, y, z, ... une indeterminee 0, et puis r^duire k C. II. 25 194 SUB LA TRANSFORMATION d'uNE FONCTION QUADRATIQUB [136 z^ro les coefficients de tous les termea dans lesquels entre cette ind^termin^ 0. Les valeurs de z, y, z, ... ne conidendront pas 0, et en repr^ntant par % rinddtermin^ que Ton doit ajouter k la suite z, y, z, ... , la valeur de % sera, comme on voit sans peine, ^ = — 0; le determinant de transformation pour la forme auz inddtermin^ w, y, z, ,..y0 sera + 1| et ce determinant sera le produit du determinant de transformation pour la forme aux ind^terminees x, y, z,.., multipli^ par —1. Le determinant de transformation pour la forme aux indeterminees x, y, z, ... sera done —1, c'est-jl-dire, la transformation sera impropre. Au lieu de la formule de transformation ci-dessus, on peut se servir des formules (f V> ?, ••) = ( h^p, b , y+x,... > ••• — 1 a, A, jr,... 9* J* c,... X*> y» *> •••)» x = 2f-a?, y = 2i;-y, z=2f-2r, .... Par exemple, en supposant que la forme k transformer soit on aura (f % t •••) = ( a, I/, -A*,... — I/, 6, X, . . . M> — ' A», c, . . . •%ax, by, cz, ...), x«2f-a:, y = 2ty-y, z = 2f--r, Ac, de mani^re qu'en posant a, v, -^,... ■ i/, 6, \, . . . M> X, c, . . . =*, oil aura (x, y, z •••)— f a, V, -/*«••• -.-1 -.'. 6, A>) ... • • • C| • • . 1 it a 2£ f 2A' 2^ // 25'-- 6' 25" 2C 2C 2(7"--... Xcue, by, cz, ...), 136] EN elle-m£mb par des substitutions lin^ires. 195 ce qui est I'^uation pour la transformation propre en elle-m^me, de la fonetion a^ + &y*-f cei* + &K^ On en d^uira, comme dans le cas gdndral, la formule pour la transformation impropre. On trouvera des observations sur eette formule dans le m^moire "Becherches ult^rieures sur les determinants gauches" [137]. Je reviens k T^uation gdn^rale ( Xx, y, z, ...)» = ( x^. y. ^. •••)*. et je suppose seulement que x, y, z, ... soient des fonctions lin&dres de x, y, z, ... qui satisfont k cette ^nation sans supposer rien davantage par rapport k la forme de la solution. Cela ^tant, je forme les fonctions lindaires z — sx, y — sy, z — sz, &c., oil s est una quantity quelconque, et je consid^re la fonetion (OXx--«a:, y-«y, z-«^ ...)», laquelle, en la d^veloppant, devient (i + «*XO)(^, y, z...y^2s(<>){x, y, Z...XI % r...); et en d^veloppant de la mSme mani^re la fonetion quadratique (0)(x--ar, y-jy, z--^, ...j , on obtient I'^uation identique / 1 1 1 \« ( OXx-«a?, y-«y, z-«^, ...)» = «».( 0)(x--a?, y--y, ^"g^'"')' Soit n 1^ determinant formd avee les coefficients de fonctions lin^ires x — sx, y-sy, z— «z, &c. En supposant que le nombre des inddtermin^es x, y, z, &c., est n, sera ^videmment une fonetion rationnelle et int^grale du degrd n par rapport k 8. Soit de mSme Q' 1^ determinant form^ avec les coefficients de 1 1 1 . x--ir, y--y, z--^> &c.; r^uation qui vient d'etre trouv^e, donne □« = «** □'«, c'est-k-dire Q = ± «* D'- Cela &it voir que les coefficients du premier et du dernier terme, du second et de Tavant- dernier terme, &c., sont dgaux, aux signes pr&& De plus, le coefficient de la plus haute puissance «* est toujours ± 1» et on voit sans peine qu'en supposant d'abord que n soit impair^ on a pour la transformation propre: n=(i, p,...p, ix-«. ir et pour la transformation impropre n = (l, -P....P, -lX-^> If: Equation qui pent 6tre chang^ en celle-ci: n = — (1> P, ... P, 1X*> I)**- P^^» ^^ supposant que n soit pair, on a pour la transformation propre: n=(i, P, ...p, ix-«, 1)^ et pour la transformation impropre: n=(i, -P....P, -ix-«» m 25—2 i 196 SUR LA TRANSFORMATION d'uNE FONCTION QUADRATIQUE [136 le coefficient moyen dtant dans ce cas 4geA k z6ro, Ces th^rfemes pour la forme du determinant des fonctions lin^aires x — «r, y — «y, z — «z,' . . . sent dus k M. Hermite. II y a ^ remarquer que la forme ( X^> y» z ...y est tout k Mi inddterminfe; c'est-^-dire, on suppose seulement que x, y, z, ... soient des fonctions lin^aires de ^> y» ^, •••> telles qu'il y ait une forme quadratique ( X^» y> ^> •••)* P<>^r laquelle r^uation ( X^, y, z ...)^ = ( X^> y, ^ ^-Y ^t satisfaite. Je regarde d'un autre point de vue ce probl^me de la transformation en elle- mSme, d'une fonction quadratique par des substitutions lin^ires. Je suppose que X, y, z, &c. soient des fonctions lin^aires donn^es de x, y, z, ... , et je cherche une fonction lin^aire de x, y, z, &c qui, par la substitution de x, y, z, &c. au lieu de X, y, z, &c. se'transforme en elle-mSme k un facteur prfea Soit (f, m, w, ...X^, y, z, ...), cette fonction lin^ire, il faut que (f, m, 7i, ...Xx, y, z, ...) soit identiquement =«.(f, m, w, ...) (a?, y, z, ...), ou, ce qui est la mfime chose, que (f, m, n, ...Xx — «p, y — «y, z — «^, ...) soit = ; c'est-ji-dire, les quantity Z, m, n, ... seront ddtermin^es par autant d'^uations lin^aires dont les coefficients sont pr^is^ment ceux de x — «a?, y — sy, z — sz, &c. ; done s sera d^termind si Ton rend dgal i zdro le determinant formd avec ces coefficients, et Z, m, n, &c. 86 trouveront donnas rationnellement en termes de 8. Cela ^tant, je suppose que les racines de T^quation en 8 soient a, 6, c, . . . , et ces diffdrentes racines correspondront aux fonctions lin&ires x^, x^, x^., ... qui ont la propri^te dont il s'agit. Soit ( X^» V* ^* •••)' une fonction quadratique qui se transforme en elle-m^me par la substitution de x, y, z, &c. au lieu de a?, y, z, &c. Cette fonction pent 6tre exprim^e en fonction quadratique de Xa, Xft, x«, &C.; quantitds qui, en substituant x, y, z, &c. au lieu de x, y, z, ... deviennent (t^H y 0X5 ) CX^ , . • » . Je prends les cas d'une fonction binairef temairej &c., et d'abord le cas d'une fonction binaire. En ^crivant d*abord ( X^> yY^i-^t ^» C^X^a> x^)*, on doit obtenir identique- ment (A, B, C) (ax«, 6xft)« = (il, B, C)(Xa, x^y, c'est-i-dire ^(a«-l) = 0, 5(a6-l) = 0, (7(6*— 1) = 0. Or la solution A=B = C = ne signifiant rien, on ne pent satisfaire k ces Equations sans supposer des relations entre les quantit^s a, 6; et pour obtenir une solution dans laquelle la fonction quadratique ne se rdduit pas k un carre, il £Etut supposer, ou aft — 1 = 0, ou a' — 1 = et 6" — 1 = 0. Le premier cas est celui de la transforination propre. II donne a6 = l, ( 0X^» y)» = ZxaX6. Le second cas est celui de la transformation impropre. II donne a = + l, 6 = -l, ( OX^* y)' = ^V-l-wXft«. En passant au cas dune fonction temaire, soit ( X^, y» ^y^(^. B, C, P, (?, jyXxa, Xft, x,)«i on doit avoir identiquement {A, B, C, F, G, HXaXa, 6x,, cx,)« = (il, «, C, F, (?, ^x„, x,, x,)«. 136] EN ELLE-mMe par DES substitutions LINilAIRES. 197 c'est-il.dire^(a«-l) = 0, £(6>-l) = 0, C(c»-1)=0, J^(6c-1) = 0, (?(ca-l) = 0, Hiah-l) = 0, et on voit que pour obtenir une solution dans laquelle la fonction quadratique ne sse r^uit pas k un carrd» ou k une fonction de deux ind^termin^es, il faut supposer par exemple a* — 1 = 0, 6c — 1 = 0. On a done dans le cas d'une fonction temaire : a'=l, 6c = 1, ( X^> y» '2:)» = Zxa' + mx6Xe. Xa transformation sera propre, ou impropre, selon que a = + l ou a = — 1. Dans le cas d'une fonction qtuitemaire, on obtient pour la transformation propre: oi = cd = 1, (<>)(a!y y, z, wy = l XaXb + m XcX^, est pour la transformation impropre: a = + l, 6 = -l, crf = l, ( X^> y. -^^ ^)' = ^ V + mxfr« + nXcXd. Dans le cas d'une fonction quinaire on obtient a*=l, 6c = dc=l, ( X^> y» ^» ^> t)' = Zx«« + mxftXc + wxrfX« est la transformation est propre ou impropre, selon que a = + 1 ou a = — 1 ; et ainsi de suite. Cette mtStbode a des difScultes dans le cas oil T^uation en ^ a des racines egales. Je n'entre pas ici dans ce sujet. Dans les formules qu'on vient de trouver, on pent consid^rer les coefficients £, m, isQ, coqime des quantit^s arbitraires. Mais en supposant que la fonction quadra- t.ique soit donnie, ces coefficients deviennent dAerminis. On les trouvera par la formule »uivante que je ne m'arrSte pas k d^montrer. Soient a, /S, 7, &c. les coefficients de la fonction lin^ire x^, a', P\ 7', &c. les ooefficients de la fonction lin^aire z*, et ainsi de suite; alors, dans les diffi^rentes fiormules qui viennent d'etre donnas, le coefficient d'un terme Xa' k droite sera -it et le coefficient d'un terme XaX^ k gauche sera (tX«» P* 7. •••X<»'> ^> 7'> •••)' oi!i k denote le discriminant de la fonction quadratique k gauche, et ou les coefficients des fonctions quadratiques des d^nominateurs sont les coefficients inverses de cette mSme foiiction quadratique k gauche ^ ' Je profite de oette oooaaion pour remarqner oonoemftnt oes reoherches que les fonnules donn^es dans ^ note stir lee foxictioiiB dn seoond ordre (t zumi. [1S4S] p. 105) [71] poor les cas de trois et de qoatre ^^tennin^es, sont ezaolea, mais que je m'^tais iromp6 dans la fonne g^n^rale dn thtor&me. [This correction U indicated toI. x. p. 589.] 198 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136 L'application de la mdthode k la forme binaire (a, b, c){m, yY donne lieu aux d^veloppements suivants. J*&;ris x=aa? + /8y, y=7« + Sy, et je repr^nte par (I, m)(x, y) ime fonetion lin^aire qui par cette substitution est transform^e en elle-mSme, au &cteur 8 pres. Nous aurons done {I, mXax + py, ya + Sy)^8 (I, m){x, y) ; r^uation pour 8 sera «»-«(S + a) + aS-)87 = 0; laquelle pent aussi 6tre ^rite comme suit: (1, -S-a, aS-/37X«, 1)' = 0. Soient 8\ tt' les racines de cette ^nation. (II est k peine n^cessaire de remarquer que «', 8"y et plus bas P, Q, sont ici ce que dans les formules gdn^rales j'ai repr^- sentd par a, 6 et Xa, x^. De mSme les Aquations p = «'«", p^8'\ p^8"\ obtenues apr^y correspondent aux ^nations a& = l, a^ = l, &* = !.) On aura «' + «" = - S-o, «V' = aS-i87, et les coefficients 2, m seront ddtennin^ rationnellement par s, Mais on pent aussi determiner ces coefficients par I'^uation I : m = ia + m7 : 1/3 +mB, qui pent etre ^crite sous la forme (/3, S-a, -.7X/, m)« = 0. et en ^liminant entre cette ^nation et Tdquation lx + my = les quantity I, m, on voit que les fonctions lin^aires Ix + my sont les facteurs de la fonetion quadratique (/3, S — a, — 7)(y, — a?)", ou, ce qui est la m^me chose, de la fonetion quadratique je repr^nte ces facteurs par P, Q et je remarque encore que T^uation en s aura des racines ^gales si (S-a)» + 4)87 = 0, et que dans ce cas, et exdttsivement dans ce cas, les fonctions P, Q ne forment qu'une seule et mSme fonetion lin^aire. Je suppose maintenant que la fonetion (a, b, c)(x, yY se transforme en elle-mSme par la substitution aa-{-fiy, yx-^By au lieu de a?, y, ou, ce qui est ici plus commode, je suppose que les deux fonctions sont ^gales k un &cteur prte, et j'^ris (a, 5, cXouv + fiy, 7a? +8y)» = p (a, 6, c)(x, yy. 136] EN ELLE-M^E PAB DES SUBSTITUTIONS LIN]6 AIRES. 199 En d^veloppant cette Equation, on obtient a;» (a, 6, cXq? - p, 2a7, 7» )^ + 2 a:y (a, 6, cXa/3, aS + fiy-p, yS )^=0. + y*(a, 6, cXy8«, 2/3S, S«-p), VoiUl trois ^nations lin^aires pour determiner par les quantitds a, /3, 7, S, consid^r^es comme donn^, les coefficients (a, b, c) de la fonction quadratique. Les coefficients de ces Equations lin^aires sont a«-p, 207, 7», flf/9, aS + fiy- p, 78, i8», 2/38, 8«-p. Le syst^me inverse par lequel on trouve les valours de a, 6, c, est «»(aS-/87)-(aS + i97 + 8")p-l-p», - jSS (aS - /87) + fl^p, - 27* (oS - /87) + 2a7p. a«8« - )8V - (S* + «») p + p*, 7* (aS — ^87) + 7"p, — 07 (aS — /97) + 7Sp, fi^(ab^fiy)^/3»p, - 2a^ (oS - /97) + 2i8Sp, a«(flfS-iS7)-(aS + /87 + a«)p + p», et le determinant, dgal^ k z6ro, donne (aS-)87-p){(aS-/97 + p)«-p(a + S)»}=0: ^uation dont les racines sont p = aS-/97, p={i(a + S)±iV(a-S)> + 4)87}>. En comparant ces valours avec cellos de s', 8'\ on voit que les racines de T^uatiou en p sont P = «V', p = «'S p = A et nous aliens voir que ces valours de p donneut en gdndral les valours F(l^ P*, Q*, pour la fonction quadratique. Soit d'abord p = aS — )87 (= ^V), et posons pour abr^ger oS — )87 — p = ^, le systfeme inverse devient: (S'-p)*-/8p. 27, -/3S^-i9p(S-a), ^<l> + /3p,2fi, -2780-p(S-a)27, (a8 + ^y-p)<^- p(S-a)», - 2a^<^ + p (S - a) 2/S, -7»^ + 7p.27, -a7<^ + 7p(S-a), (a»-p)^-7p . 2/8, et en mettant ^==0, les termes de chaque ligne (en omettant un fisK^teur) deviennent 7, ^ (5 — a), fi. On obtient ainsi dans ce cas, pour la fonction quadratique (a, b, c){x, yY la valour (% s-«, -0X^,yy> qui est en effet le produit PQ des fonctions lin^aires. n y a ^ remarquer qu'en supposant (S — a)* + 4^87 = 0, ce qui est le cas pour lequel p sera une racine triple, il n'y aura pas de changement k hire dans ce r&ultat. La fonction quadratique est, comme auparavant, le produit PQ des fonctions lin^aires; 200 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136 seulement ces deux fonctions lin^ires dans le cas actuel sont identiques, de mani^re que la fonction quadratique se r^uit k P*. Soit ensuite p = {i(a + S)±iV(a-«)» + 4i87}H=«'* ou «"»); en ^rivant p = «■ et en mettant pour abrdger a8 — ^87 — « (S + a) + «• = x» 1® systfeme inverse devient - 2iyBx - 27J? (S - 8) (S + a), {«» + « (8 + a) + aS + ffy} x + 2l3y8 (8 + a), 7^ + 7"« (8 + a), - OTx - 7« (« - «) (8 + a), /8»x + /8»« (S + a), Done, en ^rivant x = ^ ^^ ^^ omettant le fisK^teur 8{S + a), le syst^me inverse devient et les quantity dans chaque ligne sont dans le rapport l^ : hn : m*, de manifere que la fonction quadratique est dans ee cas dgale k P^ on Q^. Cela suppose que S + s ne soit ^gal k zdro. En faisant pour le moment p = 1, on en tire la conclusion qu'd. moins de supposer 8 + a = 0, il n'existe pas de fonction quadratique binaire proprement dite (fonction non carr^e) qui par la substitution impropre ax-^-fiy, 7^ + Sy pour x, y, se transforme en elle-mSme. L'^uation S-f a = donne p = aS — /97, qui est une racine double de Tdquation cubique. On remarquera en passant par rapport k la signification de r^uation S + a = 0, que Ton a en gdndral: (a, /8Xap + i9y, yx + Sy) : (7, SXflw? + i9y, ya^ + By) = (a« + i97)^ + )8(S + a)y : y(B-h a)x-{-(S' + I3y)y, et de Ik, qu'en supposant 8 + a = 0, on a (a, fiXax-Vfiy, yx + Sy) : (7, BX^x-^fiy, yx+Sy) = x, y. Cela revient k dire qu'en faisant deux fois la substitution ax + /3y, yx + Sy au lieu de X, y, on retrouve les quantity x, y, ou que la substitution est pSriodique du second ordre. II y a aussi k remarquer que dans le cas dont il s'agit, savoir pour S + a = 0, on a 8" = —8\ et que les deux fonctions lin^aires P, Q restent parfaitement dc^termindes. Nous venons de voir qu'il n'existe pas de transformation impropre d'une fonction quadratique binaire proprement dite, k moins que S + a ne soit pas = 0. Mais en supposant S + a = 0, on voit que les coefficients des ^nations pour a, b, c deviennent -fir -7(«-«X 7'> a/9, a(S-a), -a7, /8«, i9(8-a), fiy, 136] EN ELLE-M^ME PAR DES SUBSTITUTIONS UN^IAIRES. 201 c*e8t-d.-dire : les coefficients de chaque Equation sout dans le rapport de /8, 8 - a, - 7, de manifere qu'en supposant que les coefficients a, b, c satisfont k la seule Equation (a, 6, cXA S-a, -7) = 0, oil 0, i9, 7, 8 sont des quantity quelconques, telles que S + a = 0, on aura (a, 5, cXflw? + i9y, yx + Byy = -(QB- fiy)(a, h,c){x, y)». Ce n'est \k qu'un cas particulier de T^uation identique (a, 6, cXcuc + ^y, yic + Syy + (aS-fiy).{a, b, c){x, y)» = (S + a).(aa + 67, 6(S + a), 6/9 + c8) (a?, y)« +08, S-a, -7Xa, 6, c).(/8, S-a, -7Xy, -x)\ II &ut remarquer qu'en supposant toujours T^quation (a, 6, cX)8, S-a, -7) = 0, la fonction quadratique (a, 6, oX^> y)^ ^ supposant qu'elle se riduise d, un carri, est comme auparavant P* ou Q*, c'est-il-dire le carrd de Tune des fonctions lin&ires. En effet: en ^rivant (a, 6, c)(Xy yY =^ (Ix + myY, T^uation entre i, m serait ^videm- ment (/8, S — a, — 7X^» m)" = 0, de mani^re que I, m auraient les m^mes valeurs qu'au- paravant. J'ajoute que tout ce qui pr^Me par rapport k T^quation (a. by c){(xx + /9y, 7a? + Byy = p (a, 5, c^x, yY fait voir qu'^ moins que la fonction quadratique ne soit un carrel, on aura toujours p = ± (aS — ^97) ; ce qu'on savait d4jk d^s le commencement, et ce qui pent Stre ddmontrd comme a Toidinaire, en ^galant les discriminants (etc — 6") (aS — /97)" et (ac — 6")p' des deux cdt^ Je fais enfin p = 1, ce qui donne Tdquation (a, 6, cXaa? + fiy, 7a? + 8y)« = (a, 6, c^x, yY, et (en faisant abstraction du cas oil la fonction quadratique est un carr^) je tire de ce qui prdcMe les r&ultats connus, savoir, que Ton a: 1. Pour la transformation propre: aS-)87 = l, a:26:c = 7:8 — a:— /8. 2. Pour la transformation impropre: a/8 + 6(S-a)-C7 = 0. Je crois que cette discussion a iti utile pour completer la th^orie alg^brique de la forme binaire (a, 5, c){x, y)". c. n. 26 202 [137 137. RECHERCHES ULTERIEURES SUR LES DETERMINANTS GAUCHES. [From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855), pp. 299 — 313: Continuation of the Memoir t. XXXIL (1846) and t xxxviii. (1849); 5S and 69.] J'ai d6jk donn^ une formule pour le d^veloppement d'un cUterminant gauche. En prenant, pour fixer les id^es, un cas particulier, soit 12345 12345 = 11, 12, 13, 14, 15 21, 22, 23, 24, 25 31, 32, 33, 34, 35 41, 42, 43. 44, 45 51, 52, 53, 54, 55 (oii 12 = — 21, &c., tandis que les quantity 11, 22, Sec ne s'^vanoiiiasent pas), formule peut dtre ^crite comme suit: Cette 12345 12346= 11 . 22 . 33 . 44 . 55 + 11 . 22 . 33 . < :45)« + 11 . 22 . 44 . ( [35)» + 11 . , 22 . 55 . ( [34)' + 11 , oS . 44 . 1 [25y + 11 . 33 . 55 . ( [24)' + 11 , . 44 . 55 . ( [isy + 22 . 33 . 44 . ( ;i5)« + 22 , , 33 . 55 . < [uy + 22 . 44 . 55 . 1 (13)* + 33 , . 44 . 55 . ( [uy + 11 . (2345)* + 22 . (1346)' + 33 . (1245)« + 44 , , (1235)« + 55 . , (1234)'. 137] RECHEBCHBS ULTJ&RIEURES BUR LES DJ^ERMINANTS QAUCHES. 203 Lies expressions 12, 1234, <&c k droite sont ici des Pfaffians. On a 12 = 12, 1234 = 12.34 + 13.42 + 14.23 et en Aaivant encore un terme, pour mieux pr^enter la loi: 123456= 12.34.56 + 13.45.62 + 14.56.23 + 15.62.34 + 16.23.45 +12.35.64 + 13.46.25 + 14.52.36 + 15.63.42 + 16.24.53 + 12.36.45 + 13.42.56 + 14.53.62 + 15.64.23 + 16.25.34. J'ai trouvd r^mment une formule analogue pour le ddveloppement d'un dAer- fninant gauche bord^, tel que cette formule est: al234 /31234 = ; «/8, «1. a2, a3. a4 lA 11, 12. 13. 14 2/8, 21, 22, 23. 24 3A 31, 82, 33, 34 4/8, 41, . 11 42, . 22 43, . 33 44 al234 /31234 = «/9 . 4 + ay8 . 12 . 12 . 33 . 44 + a/3 . 13 . 13 . 22 . 44 + 0/8 . 14 . 14 . 22 . 33 + 0/8 . 23 . 23 . 11 . 44 + 0/8 . 24 . 24 . 11 . 33 + 0/8 . 34 . 34 . 11 . 22 + 0/8 . 1234 . 1234 + ol . /81 . 22 . 33 .44 + o2 . /82 . 11 . 33 . 44 + o3 . /83 . 11 . 22 . 44 + 04 . /84 . 11 . 22 . 33 + ol23 . /8123 . 44 + ol24 . /8124 . 33 + ol34 . /3134 . 22 + o234 . |8234 . 11. 26—2 204 RECHEBCHES ULT^IRIEURES SUB LES D^TTERMINANTS QAUCHES. [137 II est k peine n^essaire de remarqaer que dans les Pfaffians k droite, oil entrent des symboles tels que la, fil, &c, qui ne se trouvent pas dans le determinant dont il s'agit, il faut Airire la = — al, /91 = — 1/3, &c. Le symbole /8a ne se trouve ni dans le determinant, ni au cdtd droit. Cependant il convient de poser /9a = — a/9 ; car cela extant, il serait permis de transformer les P/affians, en ^rivant par exemple a/912 = — /3al2. Je remarque en passant que, si avant de poser T^uation /9a s — a/9, on suppose que les deux s}rmboles a, fi deviennent identiques (si par exemple on ^rit a = )3 = 5), on aurait par exemple a/8.12 = a/9.12 + al.2/9 + a2.i91 = 65.12 + 61.25 + 62.51 = 55.12, &c., et on retrouverait ainsi la formule pour le ddveloppement de 12346 | 12345. La nouvelle formule pent Stre appliqu^e imm^diatement au d^veloppement des determinants mineura. En effet, en se servant de la notation des matrices, on aura 11, 12, 13 21, 22, 23 31, 32, 33 — 1 123 123 + 23 23, - 13 23, - 12 32 - 23 I 13, + 13 I 13, - 21 I 31 -3'2Tl2, -STTll, +12 I 12 11. 21. 31. 41, 12. 22, 32. 42, 13, 23, 33, 43, 14 24 34 44 — 1 1234 1234 + 234 I 234, - 134 | 234, - 124 | 324, - 234 I 134, + 134 | 134, - 214 | 314, - 324 I 124, - 314 | 214, + 214 | 214, -423 I 123, - 413 | 213, -412 | 312, +123fT23 -123 1 423 -213 413 -312 1 412 et ainsi de suite. On suppose toujours que chaque terme de la matrice k droite soit divis^ par le ddnominateur commun. On voit que les determinants mineurs qui cor- respondent k des termes tels que aa, sont des determinants gauches ordinaires, avec le signe + , tandis que les determinants mineurs qui correspondent k des termes tels que a/9, sont des determinants gauches hordds tels que /9... | a..., avec le signe — . Pour donner des exemples de la verification de ces formules, je remarque que Ton doit avoir 123 123= 11 . 23 23 - 12 . 23 I 13 -13 . 32 I 12: equation qui pent aussi Stre ecrite sous la forme 123 123= 11 . 23 23 + 21 . 23 13 + 31 . 32 I 12 137] BECHEBCHES ULT^RIEURES SUB LES DJ^'ERMmANTS GAUCHES. 205 En effet, en d^veloppant les deux cdt^, on obtient: 11.22.33 + 11.(23)* + 22.(13)' + 33.(12)» = 11.(22.33 + (23)*) + 21.(21.33 + 23.13t) + 31 . (31 . 22 + 32 . 12t). On Toit que les deux termes marqu^ par un -f* se d^truisent et que I'^uation est identiqtie. On doit avoir de mSme, 1234 I 1234 = 11 . 234 - 12 . 234 - 13 . 324 - 14 . 423 ou, ce qui est la mSme chose : 1234 1234 » 11 . 234 + 21 . 234 + 31 . 324 + 41 . 423 234 134 124 123, 234 134 124 123; c'est-a-dire, en d^veloppant des deux cdt^s: 11.22.33.44 + 11.22.(34)" + 11.33.(24)' + 11.44.(23)' + 22.33.(14)' + 22.44.(13)» + 33. 44. (12)' + (1234)' = 11 [22 . 33 . 44 + 22 (34)* + 33 (42)' + 44 (23)'] + 21 [21 . 33 . 44 + 2134 . 34» + 23 . 13 . 44t + 24 . 14 . 33t] + 31 [31 . 22 . 44 + 3124 . 24* + 32 . 12 . 44t + 34 . 14 . 22t] + 41 [41 . 22 . 33 + 4123 . 23* + 42 . 12 . 33t + 43 . 13 . 22t]. Cette expression est en effet identique, comme on le voit en observant que les ^^^ termes marqu^ par un "f* se d^truisent deux k deux, et que les trois termes ntxkjrqu^ par un (*) sont ensemble ^uivalents k (1234)*. Je remarque que le nombre des termes du d^veloppement du determinant gauche ^^ toujours une puiasance de 2, et que de plus, ce nombre se r^uit k la moiti^, 6^ r^uisant k z6io un terme quelconque act. Mais outre cela, le determinant prend ^lu cette supposition la forme de determinant d'un ordre inferieur de I'unite. Je <^iksidke par exemple le determinant gauche 123 | 123. En y faisant 33 = et en Af^centoant, pour y mettre plus de clarte, tous les symboles, on trouve 123 I 123' = 11' . (23')' + 22' . (13')'. 206 RECHKRCHES ULT^RIEUBES SUB LES D^ERMHTANTB GAUCHBS. [137 De la, en ^crivant 11 = 13'. 11', 12 = 11'. 23', 22 = 18' . 22', on obtient 12 I 12 = 11 .22 + (12)' = 11'.{22'.(13')' + ll'.(23')»). 12 I 12 = 11' . 123 I 123'. c'est-ti-dire On a de mSme 1234 I 1234' = 11' . 22' . (34')» + 11' . 83' . (24')« + 22'33' (14')» + (1234')' et de 1^, en ^crivant 11 = 14'. 11', 12 = 11.24', 23 = 1234', 22 = 14'. 22', 13 = 11'. 34', 33 = 14' . 33', on obtient 123 I 123 = 11 . 22 . 83 + 11 . (28)' + 22 . (31)' + 33 (Uy = 11' . 14' {22' . 83' . (Uy + (1234')' + 11' . 22' . (34')' + 11' . 33' (24')'1, c'est-i-dire, De mdme 123 I 123 = 11' . 14' . 1234 I 1234'. 12345 I 12345' = 11' . 22' . 33' . (45')' + 11' . 22' . 44' . (SSy + 11' . 33' . 44' . (25')' + 22' . 33' . 44' . (15')» + 11' . (2346')« + 22' . (1345')* + 33' . (1245')' + 44' . (lissy. Or il est permis d'^rire 11 = 15'. 11', 12 = 11'. 25', 23 = 1235', 1234 = 2345'. 11'. 15', 22 = 15'. 22', 13 = 11'. 35', 24 = 1245', 33 = 15'. 33', 14 = 11'. 45', 34 = 1345', 44 = 15'. 44'. En effet, les quantity k gauche ne sent li^ entre elles que par la seule ^na- tion 1234 = 12.34 + 13.42 + 14.23 qui est satisfiodte identiquement par les valeurs i^ substituer pour les quantity qui y entrent. Cela dtant, on trouve d'abord: 1234 I 1234 = 11' (16')' 12345 | 12345'. 137] RBCHBBCHSS ULT^RIEUBES SUR LES DJ^TEBHINANTS OAUCHES. 207 Je prends encore on example. On a 123456 123456'= 11' . 22' . 33* . 44' . (56')» + 11' . 22' . 33' . 55' . (46')' + 11' . 22' . 44' . 55' . (36')' + 11' . 33' . 44' . 55' . (2&y + 22' . 33' . 44' . 55' . (16')' + 11' . 22' . (U56y + 11' . 33' . (2456')' + 11' . 44' . (2356')» + 11' . 55' . (2346')' + 22' . 33' . (1456')« + 22' . 44' . (1356')' + 22' . 55' . (1346')" + 33' . 44' . (1256')' + 33' . 55' . (1246')* + 44' . 65' . (1236')' + (123456')». mis d'^Tire: 11 = 16'. 11', 12 = 11'. 26', 23 = 1236', 34 = 1346'. 45 22 = 16' . 22', 13 = 11'. 36', 24 = 1246', 35 = 1356', 33 = 16'. 33', 14 = 11'. 46', 25 = 1256', 44 = 16'. 44', 15 = 11'. 66', 55 = 16'. 55', 1234 = 2346'. 11 '.16', 2345 = 128456'. 16', 1235 = 2366'. ir .16', 1245 = 2456'. 11 ' . 16'. 1345 = 3456'. 11 '.16; = 1456', car les valeun des quantity k gauche satisfont identiqaement aux ^uations qui ont lieu entre ces mSmes quantity Par exemple I'^uation 1234 = 12.34 + 13.42 + 14.23 devient 2346'. 16'= 26'. 1346' + 63'. 1246' + 46'. 1236'. Or I'expreasion k droite devient, en d^veloppant: 26' (13'. 46' + 14'. 68' + 16'. 34') + 63' (12' . 46' + 14' . 62' + 16' . 24') + 46' (12'. 36' + 13'. 62' + 16'. 28'), 208 RECHERCHES ULT^RIEUBES SUB LES D^ERMINANTS OAUCHES. [137 et les termes qui contiennent le facteur 16', donnent ensemble 16'. 2346', les autres terraes se d^truisent deux k deux. On obtient enfin, en effectuant la substitution: 12345 I 12345 » 11' . {IQ'y 123456 | 123456' ; et ainsi de suite. Je fisus les mdmes substitutions dans les matrices inverses, en supprimant cependant la derni^re ligne et la demi^re colonne de chaque matrice. On trouve ainsi, en y ajoutant les Equations ci-dessus trouv6es par rapport aux determinants: 13' . 123 I 123' 11' . 123 I 123' = 12 I 12, + 23 23', - 13 I 23' - 23 I 13', + 13 I 13' 12 12 -2 2 + 12 I 12 11 +2H, . -1|2 + lll 11' . 14' . 1234 I 1234' = 123 | 123, 14' . 1234 I 1234' + 234 I 234, - 134 | 234, - 124 | 324 -2347~i24, + 134 | 134, - 214 | 314 - 324 I 124, - 314 | 214, + 124 | 124 123 I 123 - 23 I 23 + 123 I 123 11 - 13 I 23, - 12 I 32 + 23 I 13, + 32T12, + 13 I 13, - 21 I 31 - 31 I 21, + 12 I 12 11' . (Wy . 12345 I 12345 = 1234 | 1234, 15' . 12345 I 12345' + 2345 I 2345', - 1345 | 2345', - 1245 | 3245', - 1235 | 4235' - 2345 I 1345', +1346 | 1345', - 2145 | 3145', - 2135 | 4135' - 3245 I 1245', - 3145 | 2145', +1245 | 1245', - 3125 | 4125' - 4235 I 1235', -4135 | 2135', - 4125 | 3125', + 1235 | 1235' 1234 1234 - 234 I 234 + 1234 1234 11 + 234 I 134, + 324T124, + 423 I 123, et ainsi de suite. - 134 I 234, - 124 | 324, - 123 | 423 + 134 I 134, - 214 I 314, - 213 | 413 - 314 I 214, + 124 I 124, - 312 | 423 - 413 I 213, - 412 I 312, + 123 I 123 137] BECHERCHES ULTl^IEUBES SUR LBS D^TTEBMINAITTS OAUCHES. 209 II est bon de changer un peu la forme de ces ^uationa On en d^uit sans peine 13' . 123 I 123' 2 . 23 I 23' - 123 123* 11 , - 2 . 13 I 23' - 2 . 23 I 13', + 2.13 I 13'- , 123 123' 22 12 12 -2.2| 2+ 12 I 12 11 , -2.1|2 + 2.2 1, + 2.1 1- 12 12 22 ^'.123411234' 2 . 234 234'- , 1234 1234' 11' , -2.134 I 234', -2.124 324' -2 . 234 I 134', 2 . 134 134'- , 1234 1234' 22' , -2 . 214 I 314' -2 . 324 I 124', -2.814 I 214', ..mnsi-'.^*^ 123 123 2 . 23 I 23- 123 123 11 - 2 . 13 23, + 2.23 I 13, + 2. 13 I 13- 123 123 22 + 2.32 ! 12, - 2 . 81 21, -2 ,12 32 -2 .21 31 • 123 1 123 + 2. 12 12- 33 et ainsi de suite. Les formules que je viens de presenter, peuvent 6tre appliqu^ aussitdt k la solution de la question suivante: "Trouver Xi, x,, x„ &c., fonctions lin^ires de Xi, Xi, Xj, &c. telles que 11 Xi« + 22 x,»+ 33 x," + &c. = 11 a?i" + 22 a:,» + 33 a?,« + &c./' c*est-it-dire : transformer une fonction quadratique llari' + 22ajj' + 33a:b' + &c. en elle- mSme par des substitutions lin^aires. II suffira d'^rire la solution pour le cas de trois ind^termin^: on satisfait identiquement k I'^uation 11 Xi« + 22 x,» + 33 x,« = 11 a?i» + 22 a?,* + 33 a?,» en ^rivant 1 (Xj, Xj, Xj) — 123 I 123 +2.23 I 23 123 123 11 , -2.13 I 23, -2.12 I 32 (ll«i, 22«,. 33a^). -2.23 I 13. I-2.327T2, + 2.13 I 13- 123 123 22 , -2.21 131 -2.31 I 21, +2.12 12- 123 I 123 33 C. II. 27 210 RECHEBCHE8 ULTI^EUIIES SUB LES Dl^TEBMINANTS GAUCHES. [137 Voil^ la transformation propre. On en tire la transformation impropre de ll^i' + 22d!:,'' en dle-mSme en ^rivant 33 = 0; car, cela pos^, les valeurs de Xi, x, ne contiennent pas Xg, et Ton n'a plus besoin de la valeur de x,. On obtient ainsi la solution suivante; savoir, on satis&it identiquement h T^uation en ^rivant (x,, Xj) = 123 I 123' ir Xi» + 22^ x,» = ir a?i« + 22^ a?,« 2.23 |23'-^?^jL^', -2. 13 I 23' (ira?», 22-^:,), - 2 . 23 I 13', 2 . 13 I 13 , 123 I 123^ 22 ce qui est une transformation impropre. Mais en y fiusant la substitution 11 =13'. 11', 22 = 13'. 22', 12 = 11'. 23', on r^uit la solution k celle-ci, savoir on satis&it identique- ment k r^uation llxi" + 22x,"= lla?i» + 2aci" en ^crivant (xi, ^fd- 12 I 12 1 O TO . -2.21 2 + i^^, -2.112 11 + 2 . 2|1, +2.11- 12 I 12 22 (lla^i, 2ac), ce qui est encore une transformation impropre, qui correspond de plus pr^ a la formule pour la transformation propre; la seule diffi^rence est que les signes des termes de la premiere colonne de la matrice en sent chang^ En introduisant des lettres simples a, 6, &c & la place des symboles 11, 22, &c., je consid&re d'abord la transformation propre ax* + 6y* = da^ + 6y*. Ici, en 6criyant 11. 12 ^ a, p 21. 22 -V, h la formule donne (X, y) = ab + ^ ab-i^, - 2i* 2va , ab'-i^ («. y)- La transformation impropre ax* + 6y* = ewB* + 6y* s'obtient au- moyen de la formule donnee plus bas pour la transformation propre de la fonction aa^ + bi/* + c^ en eUe-mSme, En y &srivant c = 0, on obtient (X, y) = aX« + 6^» 2\fia , — aX* + bfi* («i y)- 137] RBCHEBCHE8 ULT^RIEUIIES SUB LES D]fer£ItMINANTS GAUCHES. 211 J'ai d4jk £Edt voir de quelle mani&re cette formule se rattache k la formule pour la transformation propre; la difif<^rence entre les formes de ces transformations dans ee cas simple est assez firappante« Pour obtenir la transformation propre ax' + 6y' + cz* = cur* + 6y' + C2^, j'&ris 11. 12, 13 = 21. 22. 23 31. 32. 33 v, 6, X c eette formule donne (x, y, z) = abc + aK*-^ bfA* + ci^ a6c + aX*-6|A*-ci^, 2(\fjh-cv)b , 2(vX + 6aa)c 2(X|A+cv)a , abc — aX^ + bfjL^^cp', 2(jiv — a\)c 2 (vX — bfi) a , 2 (/LM/ + aX) 6 , a6c — aX* — 6/a' — ci^ La transformation impropre 0n eUe-mSme ax' + 6y' + cz' = oa?" + 6y* + c^' (^. y. z)' ^tre tir^ de la transformation propre en die-mime de la fonction donn^e ei-apr^s -f 6y' + C2^ + rft(;*; en y ^crivant d = 0, on obtient ^^* ^' ^^ 6cp«+ca<r» + a6T« + <^» bcp^ ■{■ cao^ •\' abi* - 4^, - 2bT(l> - 2bcp(r , 2ca<f>-2bcTp 2ar4> — 2ac/!>cr , ic/^* — caa* + air" — 0*, — 2cp<l> — 2ca<rT 2aa(l> — 2abpT , 2bp4>-2abaT , bcp* -h caa* ^ abi* — (f)^ («^» y, '^). Pour verifier que cette expression n'est en effet autre chose que la formule pour ^ transformation propre, en y changeant les signes de tous les termes, j'^ris dans la ^^Hxiule pour la transformation propre, as=& = c = a>. On a ainsi pour la transformation P^pre x' + y* + z' = a;« + y* + -8^, ^ ^nation (t. cD«+X*-/A*-i'', 2X|A-2vai , 2i^X+2^cD 2X^ + 21^0 , ®»-X* + /A«-i;*, 2^1^-2X01 2i/X-2/*® , 2/Av + 2Xa> , ««-X« + aa"-»^ (^, y, '^), 27—2 212 RECHEBCHBS ULT^BIEnitES SUB LES Dl^EBMINANTS GAUOHES. [137 et en ^rivant dans la formule pour la transformation impropre, a=& = c = l, d^^O, et \, fi, V, — m au lieu de p, cr, r, <!>, on obtient pour la transfonnation impropre r^uation x« + y«+2? = a;« + y*+-8*, a)*-X* + ^« + i^*, -2\|A+2w , -2i^X-2/*« -2X/i-2i^a) , -a)* + X*-A** + i^, 2/iav-2X» -2ifX + 2/u» , -2/iAv-2\i» , -©« + X"-At*-i^ Pour obtenir la transformation propre ax' - 6y' + cz' + rfw' = flur* + fty' + ci* + du^, 11, 12, 13, 14 21, 22, 23, 24 31, 32, 33, 34 41, 42, 43, 44 oela donne d'abord, en mettant pour abr^ger, la valeur du determinant (^» y» ^)' j'&ris a, Vf -M, P V, 6, X, <T M, -X, c, T P> -<r. -T, d — 1 1234 I 1234==aJcd + 6c/5« + ca<r* + a67* + acrx« + M/i» + c(ii;« + ^ (ce que je repr^sente par &). J'ajoute aussi la valeur de la mairicA inverse 11, 12, 13, 14 21, 22, 23, 24 31, 32, 33, 34 41, 42, 43, 44 ocd + 6t* + c/E)* +dfi*, ddK-^rp^ + dfUf^aoTt aca + fjLtf} -^ cvp +aXT, bdfi H- o-if} + dXjf — bpT , — odK ^pif) + dfjtv — aoT, abd + aa* + bf^ +dv^ , dbr + 1^^ + bfip — aXo", savoir: 1 bed +bT* +ca* +(iX«, k cdp + T^ + dXft — cpa. — Mft — cr^ + (iXi^ — bpr, bcp +X^ + ci'cr ^bfir, — 6cp — X^ + cva — 6/iT •- aca — fuf) — cpp 4-aXT — air — 1^0 + 6/Lif) — a\a abc +aX'+6/i« + ci/» 137] BBCHERCHES ULT^RIEURES SUR LES Dl^RMINANTS GAUCHES. 213 On a pour la transformation, I'^uation (x, y, z, w) = 1 k abed — 6cp' + ccur* + abi^-h adX? 2a (cdv + T0 + dXfjk — cp<r) , 2a (— bdfi — cr^ + dKv — bpr), 2a (bcp + X^ + cvtr — bfir) , 26 (— cdv — T^ + dKfi — cpcr) , a6c(i + bcp* — ccur* + abr* — ad\" 26 (adk + p^ + rf^v — a<rr) , 26 (oca + fjul> — cvp + aXr) , 2c (6d/i + cr^ + (iXv — 6pT) , 2c (— adX — p<l> + dfjkv — cutt) , a6cd + bcp^ + ccur" — a6T' — odX* - 6d^« + cdi/* - 0« 2c (a6T + 1^ + bfjLp — aXo*) , 2d (— 6cp — X0 4- ci'cr — bfir) 2d (— accr — |i^ — ci'p + aXr) 2d (— abr — i^^ + 6/ip — aXo") abed — 6c/!)' — caa* — abr* 4- adX* (or, y, J?, m;). Je suppose que Ton ait a = 6 = c = d = «, et j'&jris -^ = , c'est-i-dire ^ = — ^- -^*^ — ^ ou \p + /xo- + in- + '^a) = 0. En fiedsant cette substitution, on trouve d'abord k — fu^B, oh iJ = X« + ;i« + l;« + ^ + p« + <T« + T»+a)«, et puis pour la transformation propre x« 4- y « 4- z« + w' = a* + y ' + -8^ + W, TAjuation (x, y, z, w) = -p« + <r* + T« + tt)* + X"-^"-I^-'^, 2a>v - 2t'^ + 2X/i - 2/!)cr, - 2®/i + 2<r^ + 2Xi/ -2/yr, 2a>p - 2X^ + 2i^cr - 2/iat, 1 - 2o>v + 2t^ + 2X/i - 2pcr, 2a)X - 2p'^ + 2^1^ - 2<rr, 2a>(r - 2/i^ - 2i;/o + 2Xt, 2mp. - 2<r^ + 2Xv —2pT 2oiX + 2p^ + 2/u/- 2<rr p« + <r» - T» + «»- X«- AA« + i^ 2toT- 2i^+ 2aap - 2Xcr -^, - 2aip + 2X^ + 2va - 2/iT 2a)cr + 2^^ - 2pp + 2Xt 2ftrr + 2ir^ + 2/i/) - 2X<t -^' (a:, y, ^, w). On peut changer la forme de cette expression, en y &rivant X = i(«-aO. /t = i(/9-i8'). i' = i(7-7'). V^ = i(S-8'). 214 BECHEBCHES ULTl^RIEUBES SUR LES Dl^TEBMINANTS GAUCHES. [137 cela donne de mani^re qu'en ^crivant on obtient R = \{M-\-M')^.J{MM'), et la formule pour la transformation devient (x, y, z, w) = V {MM') aa' + i8/8' + 77' +S8', a/8'-i8a'+78'+V. cu^-fifi'+yy'+iS', aS' - ^' - yfi' - Sc^ . -»y'-y8S' + 7««'-Si8', »y' +/SS' oS' —fiy' yd - 1?, y^ + Scl. 77' + SS*, 7S' - S7'. oy + ies* + 70' + s^ afi'-ficl+y^ + Sy' ad -^ff-yy'+BS' (x. y, z, w). la Oil voit done que mdme sans supposer T^uation M = M\ cette formule donne transformation propre x« 4- y« 4. z* + w* = a;* + y* + ^ + ?*;". Cette solution est k peu pr^ de la m6me forme que la solution impropre donn^ par Euler dans son m^moire "Problema algebraicum ob affectiones prorsus singulares memorabile " Nov. Comm. Petrop., t. xv. 1770, p. 75, et Coram. Arith. collects, [4to. Petrop. 1849], t. i. p. 427. Je remarque aussi que cette ' m6me solution pent 6tre d^uite de la th^orie des Quaternions, En effet, t, j, k ^tant des quantity imaginaires telles que i*=j' = A;* = — 1, jfc = — ^' = t, ki — — ik^jy %j^''ji=sk, on obtient, en effectuant la multiplication: X, y, z, w ayant les mdmes valeurs que dans la demi^re formule de transformation. En changeant les signes des termes de la quatri&me colonne, on en tire pour la transformation impropre x' + y' + z' + w* = a:' + y* + -»• + w*. 137] RECHERCHES ULTiRIEURES SUR LES D^^TERMINANTS OAUCHES. 215 la fonnule suivante plus sym^trique : (X, y, z, w)-^^^j^^ -oa' +/9/8'+ry' + S8', -ciff -ev9'-/8a'+78'-&y', aa' -»y'-y88'-7a'+S/8', US' /9a' - 78' + 87' , aiSf -fir/ ettt'+ff^ aff-fia' To'-S/y, 77'+SS'. 7«' - V, ay - ^87' + 7)8' - So' a7' -ffS' -yc^ -SIS' afi'+fic^ -yB' -By' a«'+y8/8' + 77'-88' (a;, y, «, w). Ces formules pour la transformation, tant propre qu'impropre, de la fonction ^ + y* + '8* + W en eUe-mSme, sont utiles dans la th^rie des polygenes inscrits dans line sur£Eu» du second ordre. 216 [138 138. RECHERCHES SUR LES MATRICES DONT LES TERMES SONT DES FONCTIONS LINEAIRES D'UNE SEULE INDfiTERMINEE. [From the Jounial fiir die reine und angewandte Mathematik (Crelle), torn. L. (1855). pp. 313—317.] Je pose la matrice A y IJ I G , ... A I IJ , G , ... dont les termes {n* en nombre) sont des fonctions lin^aires d'une quantity «, et je considfere le determinant form^ avec cette matrice, et les determinants mineurs fonn& en supprimant un nombre quelconque des lignes et un nombre ^gal de colonnes de la matrice. En supprimant une sevle ligne et une aeule colonne, on obtient les premiers mineurs; en supprimant deux lignes et deux colonnes, on obtient les seconds mineurs; et ainsi de suite. Cela ^tant, je suppose que la quantity s a ^t^ trouvde en egalant k z^ro le determinant form^ avec la matrice donn^e ; ce determinant sera une fonction de s du n-i^me degre qui g^neralement ne contiendra pas de facteurs multiples. On voit done qu'un facteur simple du determinant ne pent pas entrer comme facteur dans les premiers mineurs (c'est-k-dire dans tous les premiers mineurs) ; mais en supposant que le determinant ait des facteurs multiples, un fisu^teur multiple du determinant pent entrer comme facteur (simple ou multiple) dans les premiers mineurs, ou dans les mineurs d*un ordre plus eieve. II importe de trouver le degre selon lequel un facteur multiple du determinant pent entrer comme facteur des premiers mineurs, ou des mineurs d'un ordre quelconque donne. Cela se fait trfes facilement au moyen d'une propriete generale des determinants; si les mineurs du (r4-l)ifeme ordre contiennent le facteur («— a)* (c'est-it-dire, si tous 138] BECHEBCHES SUB LES MATBICES &C. 217 les mineurs de cet ordre contiennent le facteur (« — a)*, mais non pas tous les fistcteurs (» — a)*"*"*) ; et si de m^me les mineurs du r-ifeme ordre contiennent le fiskcteur (« — a)^ ; alors les mineurs du (r — l)i&me ordre contiendront au moina le £Etcteur (« — a)^^"*. ^utrement dit: les mineurs du (r — l)ifeme ordre contiendront le facteur (« — a)^ od 'y > 2)8 — a, ou, ce qui est la mfime chose, a — 2/8 + 7 <t ; c'est-k-dire : en formant la suite des indices des puissances selon lesquelles le facteur {s — a) entre dans les Tuineurs premiers, seconds, &c. (il va sans dire que cette suite sera une suite dScroissante), lee di£f(^rences secondes seront positives [c'est-k-dire non nigativ€s\ Je repr^sente par ^<> /8, 7, ... la suite dont il s'agit; je suppose, pour fixer les iddes, que h soit le ciemier terme qui ne s'^vanouisse pas, et j'^cris a, ^, 7, S, 0, 0, . . . a-A )8-7, 7-S, S, 0,... a-2;9 + 7. i8-27 + S, 7-2S, S, 0,...; xci, quel que soit le nombre des termes, tous les nombres de la troisi^me ligne seront positi&, et en reprdsentant ces nombres par /, /', /'', &c., on obtient: a=/+2/' + 3r + 4r+..., ^= /'+2r+3r+..., 7= /-+2r+..., n y a ici k consid^rer que le nombre a, indice de la puissance selon laquelle le facteur {s — a) entre dans le determinant, est donn^ ; il sera done permis de prendre jy>ur /, /', /",... des valeurs enti^res et positives quelconques (z^ro y compris) qui aatisfont k la premiere ^nation; les autres ^nations donnent alors les valeurs de iS, % Sy &c On forme de cette mani&re une table des particularity que pent printer un facteur multiple {s — a)* du determinant ; cette table est compos^e des symboles ^> /3, y,..., et les nombres a,l3,.., de chaque symbole font voir le degr^ selon lequel le £acteur (s — a) entre dans les determinants, dans les mineurs premiers, seconds, &c. Or il est trfes facile de former, au moyen des tables pour a^l, a = 2, ... as A;, la table pour a=sit+l. On a par exemple pour a=l, a = 2, a = 3, a = 4t les tables suivantes: Pour a = 1, 1. Pour = 2, 2, 21. Pour a = 3, 3, 31, 321. Pour = 4, 4, 41, 42, 421, 4321. De \k on tire la table pour a = 5, savoir : Pour a = 6, 6, 61, 62, 621, 631, 6321, 64321. £n eifet, le premier terme est 6, et on obtient les autres termes en mettant le nombre ) devant les symboles des tables pour a^l, 0=2, a = 3, a = 4, en ayant seulement soin C. II. 28 218 RECHGRCHGS SUR LES MATRICES DONT LES TBBHES 30NT [l38 de supprimo- les symbolee 58, 54-, 541, 542, 5421 pour lesquels le premier terme de la Buite des diffdreDces eecondes est nigsAif. On trouve de mSme pour a = 6, la table suivante, savoir : Pour a = 6 6, 61, 62, 621, 63, 631, 6321, 642, 6421, 64321, 654321; et ainsi de suite. Lea nombrea dea Bymboles pour a = l, 2, 3, 4, 5, 6, 7, 8, &c sont 1, 2, 3, 5, 7, 11, 15, 22, [30, 42, 56], &c.; ce aont les coefficients des pmBSonces a^. of, a? &C. dans le d^veloppement de (1-ic)-' (l-a:')-' {l-x'y* (1-3-)-' (l-«^)-' ... &c. foQCtions qui se pr^entent, comme on sait, daos la th^rie de la pari^on des nombrea. Maintenant, au lieu de coosiddrer un seulfacteur du determinant, J e considfere toils les facteurs: par exemple pour n = 4, le determinant pout avoir un fiacteur double (s—df, et un autre facteur double (« — 6)* ; il peut de plus arriver que le facteur (« — a) soit fiwteur simple des premiers mineurs, mais que le facteur (s ~ h) n'entre paa dans lee premiers mineurs. Le eymbole qui correspond au &cteur (s — a) sera 21, et le symbole qui coirespond au facteur (B—h) sera 2. En combinaot cee deux symboles, on aura le symbole composd qui denote que le determinant a deux facteurs doubles de la clasee dont il s'agtt. Je forme de ces symboles composes n = 2, n = 3, 7i = 4, &c. On a: table* pour n = 1 , ^0' 1 2 . 21 1 1 1 _ . 31 , 321 Pour n = 4 : 1 2 21 3 321 2 21 1 1 1 1 1 1 1 1 2 2 ^ et ainsi de suite. g.g, 42 . 421 , 4321 [ 138] DES FONCTIONS LINJ&AIRES d'UNE SEULE INDlfeTERMINilE. 219 321 Pour donner encore un exemple du sens de ces symboles, le symbole denote cjue le determinant a un facteur (s — a) qui entre comme facteur triple dans le deter- minant, comme &eteur double dans les premiers mineurs, et comme facteur simple dans les seconds mineurs; Tautre facteur du determinant est un facteur simple (s — b), Les xiombres des symboles pour n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, &c. sont 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, &c. ; ces nombres sont les coefficients de oc", a?, a*, <&c, dans le developpement de ( 1 - jr)-^ (1 - aj»)-« (1 -a^)-'(l -ic*)-» (1 - a^y (1 - a:*)"" (1 - a^)-"(l - af)-^ (1 - a;»)-». . . &c. cju les indices 1, 2, 3, 5, 7, 11, &c. forment la suite qui se pr^sente dans la theorie c^e la partition des nombres, dont j'ai parie plus haut II est tr^s facile de d^montrer f^u'il en est ainsi. Les r^sultats que je viens de presenter sont en partie dus k M. Sylvester (voyez isson m^moire "An enumeration of the contacts of lines and surfaces of the second cz^rder," Philosophical Magazine, [voL I. (1851), pp. 18 — 20]). En efFet, M. Sylvester ciommence par etendre h. des fonctions d*un nombre quelconque d'indeterminees Tid^e ^^ometrique des contacts des courbes et des surfaces. £n considerant les deux ^qua- 't.ions quadratiques [7=0, F=0, il forme le discriminant de la fonction quadratique ZJ -\- aV^ et ir cherche dans quel degr^ chaque facteur de ce discriminant pent entrer c^romme facteur dans les mineurs premiers, seconds, &c. Le discriminant de M. Sylvester <=^st un determinant sym^trique ; mais cela ne change rien k la question, et je n'ai fjEiit que reproduire Tanalyse de M. Sylvester, en donnant cependant Talgorithme pour la formation des symboles, et de plus la loi pour le nombre des symboles. M. Sylvester cJonne pour n = 2, 3, 4, 6, 6, des nombres qui, en ajoutant k chacun le nombre 2, pK>ur embrasser deux cas extremes qui ne sont pas comptes, seraient 3, 6, 14, 26, 58. X\ se trouve dans le nombre 26 une erreur de calcul ; ce nombre devrait 6tre 27, et ^n suppliant le premier terme 1, on a la suite trouv^e plus haut, savoir 1, 3, 6, 14; 27, 58, &c. ; il y a de mSme une erreur de calcul dans les nombres donnas par M. Sylvester pour n = 7 et n = 8. Mais tout cela s'applique k une autre theorie g^ometrique, savoir k la theorie des figures homographes. Pour fixer les id^es, je ne considere que les figures dans le plan. Kn supposant que x, y, z soient les coordonnees d*un point, et en prenant pour (x, y, z) cles fonctions lin^aires de (ar, y, z) on aura (x, y, z) comme coordonnees d'un point honiographe au point (x, y, z). En cherchant les points qui sont homographes chacun ^ soi-mfime, on est conduit aux Equations x— «a? = 0, y — «y = 0, z — «z = 0. Les c^uantit^s k gauche x — sx, y — «y, z— sz sont des fonctions lindaires de x, y, z, ayant pour coefficients des fonctions lin^aires de s. On a ainsi une matrice dont les termes ^ont des fonctions lin^aires de s] la th^orie enti^re se rattache aux propriet^s de eette matrice. Pour le cas gdndral de Vhomographie ordinaire, on a le symbole 28—2 220 RECHERCHES SUB LES MATRICES &C. [138 1 1 1 , pour Yhomologie, le symbole ; les autres symboles 2 1 • IB- 31 se 321 au cas de Tidentit^ rapportent k des cas moins g^draux, et le symbole complete des deux figures; y compris ce cas-limite de Tidentit^ complete, il eziste pour le plan 6 espfeces d'homographie ; pour Yespace ordinaire il existe 14 especes dliomographie. Je reviendrai k cette th^rie k une autre occasion. LtmdreSy le 24 Mai 1854. 139] 221 139. AN INTRODUCTOKY MEMOIR UPON QUANTICS. [From the Philosophical Transactions of the Royal Society of London, vol. cxliv. for the year 1864, pp. 244—258. Received April 20,— Read May 4, 1854.] 1. The term Quantics is used to denote the entire subject of rational and integral fimctiens, and of the equations and loci to which these give rise ; the word " quantic " is an adjective, meaning of such a degree, but may be used substantively, the noun understood being (unless the contrary appear by the context) function; so used the word admits of the plural •' quantica" The quantities or symbols to which the expression "degree " refers, or (what is the same thing) in regard to which a function is considered as a quantic, will be spoken of as "facients." A quantic may always be considered as being, in regard to its facients, homogeneous, since to render it so, it is only necessary to introduce as a facient unity, or some symbol which is to be ultimately replaced by unity; and in the cases in which the facients are considered as forming two or more distinct sets, the quantic may, in like manner, be considered as homogeneous in regard to each set separately. 2. The expression "an equation," used without explanation, is to be understood as meaning the equation obtained by putting any quantic equal to zero. I make no absolute distinction between the words "degree" and "order" as applied to an equation or system of equations, but I shall in general speak of the order rather than the degree. The equations of a system may be independent, or there may exist relations of connexion between the different equations of the system; the subject of a system of equations so connected together is one of extreme complexity and difficulty. It will be sufficient to notice here, that in any system whatever of equations, assuming only that the equations are not more than sufficient to determine the ratios of the facients, and joining to the system so many linear equations between the facients as will render the ratios of the facients determinate, the order of the system is the same thing as the order of the equation which determines any one of these ratios; it is clear that for a single equation the order so determined is nothing else than the order of the equation. 222 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39 3. An equation or system of equations represents, or is represented by a locus. This assumes that the facients depend upon quantities x^ yi ••• the coordinates of a point in space ; the entire series of points, the coordinates of which satisfy the equation or system of equations, constitutes the locus. To avoid complexity, it is proper to take the facients themselves as coordinates, or at all events to consider these facients as linear functions of the coordinates; this being the case, the order of the locus will be the order of the equation, or system of equations. 4. I have spoken of the coordinates of a point in space, I consider that there is an ideal space of any number of dimensions, but of course, in the ordinary acceptation of the word, space is of three dimensions; however, the plane (the space of ordinary plane geometry) is a space of two dimensions, and we may consider the line as a space of one dimension. I do not, it should be observed, say that the only idea which can be formed of a space of two dimensions is the plane, or the only idea which can be formed of space of one dimension is the line ; this is not the case. To avoid complexity, I will take the case of plane geometry rather than geometry of three dimensions; it will be unnecessary to speak of space, or of the number of its dimensions, or of the plane, since we are only concerned with space of two dimensions, viz. the plane ; I say, therefore, simply that x, y, z are the coordinates of a point (strictly speaking, it is the ratios of these quantities which are the coordinates, and the quantities x, y, z themselves are indeterminates, i.e. they are only determinate to a common factor prhs, so that in assuming that the coordinates of a point are a, )8, 7, we mean only that x : y : z^a : 0:y, and we never as a result obtain x, y, z = a, /3, 7, but only x : y : z^a : /3 : y; but this being once understood, there is no objection to speaking of x, jf, z as coordinates). Now the notions of coordinates and of a point are merely relative; we may, if we please, consider x : y : z as the parameters of a curve containii^ two variable para- meters; such curve becomes of course determinate when we assume x : y : z^a : fi : y, and this very curve is nothing else than the point whose coordinates are a, 0, 7, or as we may for shortness call it, the point (a, /3, 7). And if the coordinates (x, y, z) are connected by an equation, then giving to these coordinates the entire system of values which satisfy the equation, the locus of the points corresponding to these values is the locus representing or represented by the equation ; this of course fixes the notion of a curve of any order, and in particular the notion of a line as the curve of the first order. The theory includes, as a very particular case, the ordinary theory of reciprocity in plane geometry; we have only to say that the word "point" shall mean "line,* and the word "line" shall mean "point," and that expressions properly or primarily applicable to a point and a line respectively shall be construed to apply to a line and a point respectively, and any theorem (assumed of course to be a purely descriptive one) relating to points and lines will become a corresponding theorem relating to lines and points; and similarly with regard to curves of a higher order, when the ideas of reciprodtv applicable to these curves are properly developed. 5. A quantic of the degrees m, m'... in the sets (a?, y...), {x\ y'...) &a will for the most part be represented by a notation such as m m' (♦$^, y...$a:', y'... )...). 139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 223 ^where the mark * may be considered as indicative of the absolute generality of the qaantic; any such quantic may of course be considered as the sum of a series of therms afy^...fl/*y^..., &c. of the proper degrees in the different sets respectively, each -term multiplied by a coefHcient ; these coefficients may be mere numerical multiples of single letters or elements such as a, b, c,..., or else functions (in general rational siod integral functions) of such elements ; this explains the meaning of the expression *' the elements of a quantic": in the case where the coefficients are mere numerical xnnltiples of the elements, we may in general speak indifferently of the elements, or of the coefficient& I have said that the coefficients may be numerical multiples of single letters or elements such as a, b, c, ...; by the appropriate numerical coefficient of a term afy^...a:'*y^ ..., I mean the coefficient of this term in the expansion of m (a: + y...) (a/ + y'... )...); ^nd I represent by the notation m m (a, b,...^x, y,...$J?', /,...)...)> quantic in which each term is multiplied as well by its appropriate numerical coeffi- cient as by the literal coefficient or element which belongs to it in the set (a, 6,...) of literal coefficients or elements. On the other hand, I represent by the notation m m (tt, b,...\x, y,...^a?', /,...)...)» quantic in which each term is multiplied only by the literal coefficient or element "vvhich belongs to it in the set (a, &,...) of literal coefficients or elements. And a like distinction applies to the case where the coefficients are functions of the elements (a, 6, ...). 6. I consider now the quantic « m' ^Knd selecting any two facients of the same set, e.g. the facients x, y, I remark that t^here is always an operation upon the elements, tantamount as regards the quantic t:o the operation ady; viz. if we differentiate with respect to each element, multiply V)y proper functions of. the elements and add, we obtain the same result as by differ- entiating with 3y and multiplying by x. The simplest example will show this as "^rell as a formal proof; for instance, as regards Saa^ -^ bxy -\- 5cy^ (the numerical Coefficients are taken haphazard), we have ^a + lOcd^ tantamount to xdy', as regards c^ (a: — ay) (^ — i8y ), we have — a(a+/8)3a + a^«4-j8^^ tantamount to ady, and so in any cither case. I represent by [ady] the operation upon the elements tantamount to aSy, s^nd I write down the series of operations 'Where x, y are considered as being successively replaced by every permutation of two clifferent £Eu;ients of the set (a:, y,...); ^> }/ as successively replaced by every permutation of two different facients of the set {x\ y',...)> fti^d so on; this I call an entire system, and 224 AN INTEODCJCTORY MEMOIB UPON QUANTIC8. [l39 I say that it is made up of partial systems ooiresponding to the different fiunent sets respectively; it is clear from the definition that the quantic is redaoed to zero by each of the operations of the entire system. Now, besides the quantic itself, there are a variety of other functions which are reduced to zero by each of the operations of the entire system; any such function is said to be a oovariant of the quantic, and in the particular case in which it contains only the elements, an invariant. (It would be allowable to define as a covariant quoad any set or sets^ a function which is reduced to zero by each of the operations of the corresponding partial system or systems, but this is a point upon which it is not at present necessary to dwell) 7. The definition of a covariant may however be generalized in two directions: we may instead of a single quantic consider two or more quantics; the operations {aSy], although represented by means of the same symbols x, y have, as regards the different quantics, different meanings, and we may form the sum 2 {^y}> where the summation refers to the different quantics: we have only to oondder in place of the system before spoken of, the system 2{a3y} — aj9y, ... ; 2{a:'3y'} —x'd^y ... &c. &c., and we obtain the definition of a covariant of two or more quantics. Again, we may consider in connexion with each set of £Eu;ients any number of new sets, the facients in any one of these new sets corresponding each to each with those of the original set; and we may admit these new sets into the covariant. This gives rise to a sum 8[a!d^y where the summation refers to the entire series of cor- responding sets. We have in place of the system spoken of in the original definition, to consider the system [ocdy] '-S{aSy\ ... [x'd^] — 8 {afd^\ ... &c. Ac., or if we are dealing with two or more quantics, then the system 2 [xdy] — S{ady), ... ; 2 {^'Sy'} '-S(x'd^), ... &c. &c., and we obtain the generalized definition of a covariant. 8. A covariant has been defined simply as a function reduced to zero by each oi the operations of the entire system. But in dealing with given quantics, we may without loss of generality consider the covariant as a function of the like form with the quantic, Le. as being a rational and integral function homogeneous in regard to the different sets separately, and as being also a rational and integral function of the elements. In particular in the case where the coefficients are mere numerical multi- ples of the elements, the covariant is to be considered as a rational and integral function homogeneous in regard to the different sets separately, and also homogeneous in regard to the coefficients or elementa And the term "covariant" includes, as already remarked, "invariant." It is proper to remark, that if the same quantic be represented by means of different sets of elements, then the symbols {xdy} which correspond to these different forms 139J AN INTRODUCTORY MEMOIR UPON QUANTICS. 225 of the same quantic. are mere transformatioDs of each other, i.e. they become in virtue of the relations between the different sets of elements identical. 9. What precedes is a return to and generalization of the method employed in the iirst part of the memoir published in the Camh. Math, Jour,, t, iv. [1845], and Camb. a,nd DiM. Math. Jour., t, I. [1846], under the title "On Linear Transformations," [13 WLnd 14], and Crelle, t. xxx. [1846], under the title "M^moire sur les Hyperd^termi- iiants," [*16], and which I shall refer to as my original memoir. I there consider in fact the invariants of a quantic linear in regard to n sets each of them of m facients, and I represent the coefEcients of a term Wrj/tZt... by rst...; there is no diflSculty in seeing that a, 13 being any two <lifferent numbers out of the series 1, 2, ...m, the operation {^/sda;J is identical with the operation ^S... last,., -jr^, ^%vhere the summations refer to «, ty... which pass respectively from 1 to m, both inclu- ^ve; and the condition that a function, assumed to be an invariant, i.e. to contain *jnly the coeflScients, may be reduced to zero by the operation {«/i9ajJ — a?^aj«» is of «30urse simply the condition that such function may be reduced to zero by the opera- tion {^/i^xj ; ^he condition in question is therefore the same thing as the equation cDf my original memoir. 10. But the definition in the present memoir includes also the method made use CDf in the second part of my original memoir. This method is substantially as follows: ^isonsider for simplicity a quantic £/' = ^i^ntaining only the single set {x, y...), and let CT,, ^, ... be what the quantic becomes "Vrhen the set {x^y ...) is successively replaced by the sets (a:,, yi, ...), (ar„ yj,...), ... the Xiumber of these new sets being equal to or greater than the number of facients in t;he set. Suppose that A, B, C7, ... are any of the determinants ^«ii ^af|> ^«,> ••• 9yi> 9y»> 9y,> ^hen forming the derivative '^here p, q, r ... are any positive integers, the function so obtained is a covariant in- xrolving the sets (a?i, yi,...)> (^f> yf>«««) ^m ^^^ if *ft®r *^h® diflferentiations we replace c. n. 29 226 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39 these sets by the original set {x, y, ...)* ^^ hAwe a co variant involving only the original set (x, y, ...) and of course the coefficients of the quantic. It is in £act easy to show that any such derivative is a covariant according to the definition given in this Memoir. But to do this some preliminary explanations are necessary. 11. I consider any two operations P, Q, involving each or either of them differeD- tiations in respect of variables contained in the other of them. It is required to investigate the effect of the operation P . Q, where the operation Q is to be in the first place performed upon some operand tl, and the operation P is then to be per- formed on the operand Qfl. Suppose that P involves the differentiations da, ^,... in respect of variables a, b, ... contained in Q and il, we must as usual in the operatioD P replace da, dj,,..- by da + d'a, 96 + d^6> ••• where the unaccentuated symbols operate only upon il, and the accentuated symbols operate only upon Q. Suppose that P is expanded in ascending powers of the symbols 3^., 9'6, ...f viz. in the form P-f-Pj + Pi + Ac., we have first to find the values of PjQ, PtQ, &c., by actually performing upon Q as operand the differentiations 9'a, 9'6--«- The symbols PQ, P,Q, P,Q, &c. will then contain only the differentiations d^, d^, ... which operate upon il, and the meaning of the ex- pression being once understood, we may write P.Q = PQ + P,Q + P,Q + &c. In particular if P be a linear function of 9a, dt, ..., we have to replace P by P + Pi, where P, is the same function of d'a, S't. ••• that P is of 3., 96, ..., and it is therefore clear that we have in this case P.Q = PQ + P(Q). where on the right-hand side in the term PQ the differentiations 9., 96,... are con- sidered as not in an}rwise affecting the symbol Q, while in the term P(Q) these differentiations, or what is the same thing, the operation P, is considered to be per- formed upon Q as operand. Again, if Q be a linear function of a, 6, c, ..., then PiQ = 0, PiQ = 0, Ac, and therefore P.Q = PQ-^PiQ\ and I shall in this case also (and consequently whenever P,Q = 0, P,Q = 0, &c.) write P.Q^PQ-^-PiQl where on the right-hand side in the term PQ the differentiations 9a, 96,... are con- sidered as not in anywise affecting the symbol Q, while the term P(Q) is in each case what has been in the first instance represented by PiQ. We have in like manner, if Q be a linear function of 9a, 96, 9c, ..., or if P be a linear function of a, 6, c, . . . , Q.P = QP + QiP); and from the two equations (since obviously PQ^QP) we derive P.Q-Q.P = P(Q)-Q(P), which is the form in which the equations are most frequently useful. 139] AN INTRODUCTORY MEMOIR UPON QU ANTICS. 227 12. I return to the expression and I suppose that after the dififerentiations the sets (or,, yi, ...), (^9, ys> •••)> ^^' ^^^ replaced by the original set (x, y, ...). To show that the result is a co variant, we must prove that it is reduced to zero by an operation 39 = It is easy to see that the change of the sets (x^, yi, ...), (x^ yji •••)» ^^ i^^ ^^^ original set (Xf y, ...) may be deferred until after the operation iD, provided that aSy is replaced by ^9y, + ^y,4-..., or if we please by Sxdy; we must therefore write ^ = {xidy} — Sxdy. Now in the equation il.B-B.il=il(B)-B(il), where, as before, A (W) denotes the result of the operation A performed upon iB as operand, and similarly ^(A) the result of the operation 39 performed upon A as operand, we see first that A (W) is a determinant two of the lines of which are identical, it is therefore equal to zero; and next, since 39 does not involve any differentiations affecting A, that 39 (A) is also equal to zero. Hence il . 39 ~ 39 . il = or A and 39 are convertible. But in like manner 39 is convertible with B, 0, &c., and consequently B is convertible with A^B^C^.... Now 391/itr,... =0; hence m.APB^C'...L\U,...^0, or ApB^O"... I/it/,... is a covariant, the proposition which was to be proved. 13. I pass to a theorem which leads to another method of finding the covariants of a quantic. For this purpose I consider the quantic m m (♦$«?, y...\af, y'.. .)...), the coeflScients of which are mere numerical multiples of the elements (a, 6, c, ...); and in connexion with this quantic I consider the linear functions ^x-\-r)y..., fa/ + i;y"«> which treating (f, 17,...), (f, V»'«-)> *c- ®* coeflScients, may be represented in the form (f, 17, ...$ar, y, ...\ (f , 17', ...$a?', y', ...),... we may from the quantic (which for convenience I call U) form an operative quantic (♦$f, 17,... $f, V,...)---) (I call this quantic 6)» the coeflScients of which are mere numerical multiples of da, dt, 9«, •••> and which is such that ie. a product of powers of the linear functions. And it is to be remarked that as regards the quantic 6 and its covariants or other derivatives, the symbols da, df,, da... are to be considered as elements with respect to which we may differentiate, &c. 29—2 228 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39 The quantic B gives rise to the symbols {^^], &c. analogous to the symbols {^y|, &c. formed from the quantic U. Suppose now that 4> is any quantic containing as well the coefficients as all or any of the sets of 6. Then [xdy] being a linear function of a, b, Ci... the variables to which the differentiations in ^ relate, we have again, [rfi^] being a linear function of the differentiations with respect to the variables^ da, dtt dc,... in 4>, we have these equations serve to show the meaning of the notations ^({^y}) and {n9(} (4>X an there exists between these symbols the singular equation 14. The general demonstration of this equation presents no real difficulty, but toKn avoid the necessity of fixing upon a notation to distinguish the coefficients of the different terms and for the sake of simplicity, I shall merely exhibit by an exampl the principle of such general demonstration. Consider the quantic Cr= flur» + 36a;"^ + 3cy» + dy», this gives = pda + ^hfii, + ^rfde + rfd^ ; or if, for greater clearness, da, dt, 9c, da are represented by a, fiy 7, S, then and we have {aSy] = 369a + 2c96 4- dde, and {nd(} = 3a9^ + 2^89^ 4- 79a. Now considering 4> as a function of 9a, 9^, 9e, da, or, what is the same thing, ol a, A 7, S, we may write 4> {{xdy}) = a> (36a + 2c/8 + c^) ; and if in the expression of <P we write a + 9a, ff-^-d^, 7 + 9c, 8 + 9^ for a, /8, 7, S (whe only the symbols 9a, d^, de, da are to be considered as affecting a, 6, c, ct as contain in the operand 36a + 2cl3 + dy), and reject the first term (or term independent o 9a. 9^, 9c, 9rf in the expansion) we have the required value of 4>({d?Sy)}. This value is (9a4>9a + 9^4>96 + 9ya>9c)(36a + 2c/8 + *y); performing the differentiations 9a, 9^, de, da, the value is (3a9^ + 2/89y 4- 79«) *, i.e. we have * ({^y}) = {^«} (*)• 139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 229 15. Suppose now that 4> is a covariant of 8, then the operation <I> performed upon any covariant of TJ gives rise to a covariant of the system (f, 17, ...$a:, y, ...), (f, V» •••$^» y'» •••)» &c. To prove this it is to be in the first instance noticed, that as regards (f, 17, ...$a?, y, ...), &c. we have [aody] = 178^, &c. Hence considering [xdy], &c. as referring to the quantic U, the operation 2 {aj0y} — aidy will be equivalent to [aidy] + 178^ — aSy, and therefore every covariant of the s)rstem must be reduced to zero by each of the operations iB = [xdy] + 179^ - a^y. This being the case, we have ID . ^ = B* 4- ID (^), ^nations which it is obvious may be replaced by ^ind consequently (in virtue of the theorem) by B.^ = B4> + ^3f(^), a>.iD = ^B + {i73f}(4>); ^nd we have therefore B . a> - 4>.iD = - ({179^} - 173^)(4>) ; or, since 4> is a covariant of 0, we have iD . 4> = 4> . iD. And since every covariant of the system is reduced to zero by the operation 39, and therefore by the operation <>.19, such covariant will also be reduced to zero by the operation iD.<I>, or what is the same thing, the covariant operated on by 4>, is reduced to zero by the operation "9 and is therefore a covariant, i.e. <I> operating upon a covariant gives a covariant. 16. In the case of a quantic such as {7 = (♦i^> y$^'» yO--X ^e may instead of the new sets (f, ^), (f, rf)... employ the sets (y, — a?), (y', -a?')» &c- The operative quantic 8 is in this case defined by the equation 0Cr=O, and if 4> l)e, as before, any covariant of %, then 4> operating upon a covariant of U will give « covariant of U. The proof is nearly the same as in the preceding case ; we have instead of the equation ^({^y})= {179^} (*) the analogous equation Avhere on the left-hand side [aody] refers to TJy but on the right-hand side [acdy] refers to 0, and instead of ID = {aSy} 4- ^^ — aSy we have simply ID = {aSy} — aSy . 230 AN INTRODUCTORY MEMOIR UPON QUANTICS. [139 17. I pass next to the quantic which I shall in general consider under the form (a, b,...b\ a^'^x, y)"», but sometimes under the form (a, 6, ...6\ a'^x, y)** the former notation denoting, it will be remembered, and the latter notation But in particular cases the coefficients will be represented all of them by unacceu- tuated letters, thus (a, 6, c, dP^x, y)* will be used to denote CM;* + 36a;^-|-3ca:y* + (iy', and (a, 6, c, d^x, yY will be used to denote cwc* + fta;^ + cjcy* -f dy*, and so in all similar cases. Applying the general methods to the quantic (a, 6, ...6\ a'$a?, y)"*, we see that {y^*} = a96+ 269e...+m6'3a*, in fact, with these meanings of the symbols the quantic is reduced to zero by each of the operations {yS*} — y9«, {a;9y} — xdy ; hence according to the definition any function which is reduced to zero by each of the last-mentioned operations is a covariant of the quantic. But in accordance with a preceding remark, the covariant may be con- sidered as a rational and integral function, separately homogeneous in regard to the &cients {x, y) and the coefficients (a, 6, ...&\ a^). If instead of the single set (a;, y) the covariant contains the sets (^, y^), (^, y,), &c., then it must be reduced to zero by each of the operations ly9»} — /Sy9aj, {aSy}— /Sady (where Sy3,B = yi3,B, -fyiS^-l- ...), but I shall principally attend to the case in which the covariant contains only the set Suppose, for shortness, that the quantic is represented by U^ and let CTi, IT,,... be what U becomes when the set (x, y) is successively replaced by the sets {xi^ y^), (^i y%)i ^ Suppose moreover that 12 = 8^^8,^ — d^^dy,, &c., then the function 12P13«23^... CTjlTjCr,..., in which, after the differentiations, the new sets {x^ yi), (a^, y^^... may be replaced by the original set (x^ y), will be a covariant of the quantic {7. And if the number 39] AN INTRODUCTORY MEMOIR UPON QUANTIOS, 231 f differentiations be such as to make the facients disappear, ie. if the sum of all he indices p, 9,... of the terms 12, &c. which contain the symbolic number 1, the am of all the indices p, r, ... of the terms which contain the symbolic number 2, nd so on, be severally equal to the degree of the quantic, we have an invariant, lie operative quantic becomes in the case under consideration he signs being alternately positive and negative; in fact it is easy to verify that this xpression gives identically OCT^O, and any co variant of operating on a covariant •f U gives rise to a covariant of U, 18. But the quantic (a, 6, . . . b\ a'^x, y)*", onsidered as decomposable into linear factors, ie. as expressible in the form a(a?-ay)(a?-^y)..., [ives rise to a fresh series of results. We have in this case {y3*}= 3.4- dp..., {xdy} = - (a + fi...) ada + a«. + /ff©^ + ... ; n fact with these meanings of the symbc4s the quantic is reduced to zero by each )f the operations {ady} — aSy, {y9«} — y3aj, and we have consequently the definition of he covariant of a quantic considered as expressed in the form a(a? — ay)(a? — /3y),... \jid it will be remembered that these and the former values of the symbols [xdy] and ydg] are, when the same quantic is considered as represented under the two forms a, 6, . . .i\ a'$a?, yy* and a (a? — ay) (a? — I3y). . . , identical 19. Consider now the expression a« (^ - ayy (a: - i8y)*...(a - i8)P... , ¥here the sum of the indices j, p,.., of all the simple &ctors which contain a, the lum of the indices k, p,... of all the simple factors which contain fi, Sec are respec- ively equal to the index of the coefficient a. The index d and the indices p, &c. nay be considered as arbitrary, nevertheless within such limits as will give positive ralues (0 inclusive) for the indices j, k,.,,. The expression in question is reduced to zero by each of the operations xdy] — xdy, [yds] — yd^ ; and this is of course also the case with the expressions )btained by interchanging in any manner the roots a, fi, 7,..., and therefore with the expression a«2(a?-ay)i(a?-/3y)»...(a-^)p..., where 2 denotes a summation with respect to all the different permutations of the roots a, fi, ... . 232 AN INTEODUCrrORY MEMOIR UPON QU ANTICS. [139 The function so obtained (which is of course a rational function of (a, 6, ...V, a)) will be a covariant, and if we suppose /i = md — 25/>, where Sp denotes the sum of all the indices p of the different terms (a — fiy, &c., then the covariant will be of the order fi (Le. of the degree fi in the facients x^ y), and of the degree in the co- efficienta 20. In connexion with this covariant a* 2 (a? ~ ay)P (a; - i8y)*... (a - /3)P..., of the order fi and of the degree in the coefHcients, of the quantic C/'= a(a?~ay)(a?-^y)..., consider the covariant of a quantic F= in. which, after the differentiations, the sets (xi, yi), (x^, yj,), ... are replaced by the original set (a?, y). The last-mentioned covariant will be of the order m (^ — ^) 4- /*, and will be of the degree m in the coeflScients; and in particular if ^ = 5, i.e. if V be a quantic of the order d, then the covariant will be of the order fA and of the degree m in the coefficients. Hence to a covariant of the degree in the coefficients, of a quantic of the order m, there corresponds a covariant of the degree m in the coefficients, of a quantic of the order 0] the two covariants in question being each of them of the same order /t. And it is proper to notice, that if we had commenced with the covariant of the quantic F, a reverse process would have led to the covariant of the quantic U. We may, therefore, say that the covariants of a given order and of the degree in the coefficients, of a quantic of the order m, correspond each to each with the covariants of the same order and of the degree m in the coefficients, of a quantic of the order 0; and in particular the invariants of the degree d of a quantic of the order m, correspond each to each with the invariants of the degree ni of a quantic of the order 0. This is the law of reciprocity demonstrated by M. Hermite, by a method which (I am inclined to think) is substantially identical with that here made use of, although presented in a very different form: the dis- covery of the law, considered as a law relating to the number of invariants, is due to Mr Sylvester. The precise meaning of the law, in the last-mentioned point of view, requires some explanation. Suppose that we know all the really independent invariants of a quantic of the order m, the law gives the number of invariants of the degree m of a quantic of the order (it is convenient to assume > m), viz. of the invariants of the degree in question, which are linearly independent, or asyzygetic, Le. such that there do not exist any merely numerical multiples of these invariants having the sum zero; but the invariants in question may and in general will be connected inter se and with the other invariants of the quantic to which they belong by non-linear equations : and in particular the sjrstem of invariants of the degree m will comprise all the invariants of that degree (if any) which are rational and integral 139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 233 functions of the invariants of lower degrees. The like observations apply to the system of covariants of a given order and of the degree m in the coefficients, of a quantic of the order ft 21. The number of the really independent covariants of a quantic (♦$«?, y)"* is precisely equal to the order m of the quantic, Le. any covariant is a function (generally an irrational function only expressible as the root of an equation) of any m independent covariants, and in like manner the number of really independent in- variants is m — 2 ; we may, if we please, take m — 2 really independent invariants as part of the system of the m independent covariants; the quantic itself may be taken as one of the other two covariants, and any other covariant as the other of the two covariants; we may therefore say that every covariant is a function (generally an irrational function only expressible as the root of an equation) of m — 2 invariants, of the quantic itself and of a given covariant. 22. Consider any covariant of the quantic (a, 6, ... b\ a'Jx, yf, and let this be of the order /i, and of the degree in the coefficients. It is very easily shown that md — fi is necessarily even. In particular in the case of an invariant (Le. when /t = 0) m0 is necessarily even^: so that a quantic of an odd order admits only of invariants of an even degree. But there is an important distinction between the cases of md^fi evenly even and oddly even. In the former case the covariant remains unaltered by the substitution of (y, x), (a\ b\ ... b, a) for {x, y), (a, b, ... 6\ a'); in the latter case the effect of the substitution is to change the sign of the covariant. The covariant may in the former case be called a symmetric covariant, and in the latter case a skew covariant. It may be noticed in passing, that the simplest skew invariant is M. Hermit e*s invariant of the degree 18 of a quantic of the order 5. 23. There is another very simple condition which is satisfied by every covariant of the quantic (a, b,...b\ a^x, y)*", viz. if we consider the facients (x, y) as being respectively of the weights i, — i, and the coefficients (a, 6, ...6\ a) as being respectively of the weights — ^m, — ^+1, ...^m — 1, ^m, then the weight of each term of the covariant will be zero. This is the most elegant statement of the law, but to avoid negative quantities, the state- ment may be modified as follows: — if the facients (x, y) are considered as being of the weights 1, respectively, and the coefficients (a, b...V, a') as being of the weights 0, 1, ...,m — 1, m respectively, then the weight of each term of the covariant will be i(wi^ + /*)- ^ I may remark that it was only M. Hermite^s important discovery of an invariant of the degree 18 of a qnantio of the order 5, which removed an erroneous impression which I had been under from the oom- mencement of the subject, that mO was of necessity evenly even. C. II. 80 234 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39 24. The preceding laws as to the form of a covariant have been stated here by way of anticipation, principally for the sake of the remark, that they so tso' define the form of a covariant as to render it in very many cases practicable with a moderate amount of labour to complete the investigations by means of the operation {ady} — xd^ and {ydg] — yd^^ In fact, for finding the covariants of a given order, and of a given degree in the coefficients, we may form the most general function of the proper order and degree in the coefficients, satisfying the prescribed conditions as to symmetry and weight: such function, if reduced to zero by one of the operations in question, will, on accoimt of the symmetry, be reduced to zero by the other of the operations in question; it is therefore only necessary to effect upon it, e.g. the operation {ody} — a:9y, and to determine if possible the indeterminate coefficients in such manner as to render the result identically zero: of course when this cannot be done there is not any covariant of the form in question. It is moreover proper to remark, as regards invariants, that if an invariant be expanded in a series of ascending powers of the first coefficient a, and the first term of the expansion is known, all the remaining terms can be at once deduced by mere differentiations. There is one very important case in which the value of such first term (i.e. the value of the invariant when a is put equal to 0) can be deduced firom the corresponding invariant of a quantic of the next inferior order; the case in question is that of the discriminant (or function which equated to zero expresses the equality of a pair of roots); for by Joachimsthars theorem, if in the discriminant of the quantic (a, b, ... 6\ a^$a?, y)^ we write a = 0, the result contains 6* as a factor, and divested of this factor is precisely the discriminant of the quantic of the order m — 1 obtained firom the given quantic by writing a = and throwing out the &ctor x: this is in practice a very convenient method for the calculation of the discriminants of quantics of successive orders. It is also to be noticed as regards covariants, that when the first or last coefficient of any covariant (i.e. the coefficient of the highest power of either of the facients) is known, all the other coefficients can be deduced by mere differentiations. Postscript added October 7th, 1854. — I have, since the preceding memoir was written, found with respect to the covariants of a quantic (♦ $a7, y)"*, that a function of any order and degree in the coefficients satisfying the necessary condition as to weight, and such that it is reduced to zero by one of the operations {xdy} — xdy, [ydxl—ydzf will of necessity be reduced to zero by the other of the two operations, i.e. it will be a covariant; and I have been thereby led to the discovery of the law for the number of asyzygetic covariants of a given order and degree in the coefficients; from this law I deduce as a corollary, the law of reciprocity of MM. Sylvester and Hermite. I hope to return to the subject in a subsequent memoir. 140] 235 140. RESEARCHES ON THE PARTITION OF NUMBERS. [From the Philosophical Transactions of the Royal Society of London, vol. cxlv for the year 1855, pp. 127—140. Received April 14,— Read May 24, 1855.] I PROPOSE to discuss the following problem : " To find in how many ways a number q can be made up of the elements a, 6, c, ... each element being repeatable an indefinite number of times." The required number of partitions is represented by the notation P(a, 6, c, ...)?, and we have, as is well known, P{a,b.c, ...)3 = coefficient afl in (- i_^)(i _^)(i _^)... . where the expansion is to be effected in ascending powers of x. It may be as well to remark that each element is to be considered as a separate and distinct element, notwithstanding any equalities which may exist between the numbers a, 6, c, . . . ; thus, although a = 6, yet a -f a + a -h &c. and a 4- a -f 6 -f &c. are to be considered as two different partitions of the number q, and so in all similar cases. The solution of the problem is thus seen to depend upon the theory, to which I now proceed, of the expansion of algebraical firactions. Consider an algebraical fraction v , jx where the denominator is the product of any number of factors (the same or different) of the form 1 — a^**. Suppose in general that [1 — a^] denotes the irreducible factor of 1 — af^, L e. the factor which, equated to zero, gives the prime roots of the equation 1 - a:^ = 0. We have 30—2 236 RE8EARCHBS ON THE PARTITION OF NUMBEB& [l40 where m' denotes any divisor whatever of m (unity and the number m itself not excluded). Hence, if a represent a divisor of one or more of the indices m, and h be the number of the indices of which a is a divisor, we have /a? «n[l -«•]*. Now considering apart from the others one of the multiple £EU^rs [! — «•]*, we may write /a: = [1 - «•]*/«• ^ is decomposed 4- Ac, where I{x) denotes the integral part, and the &c. refers to the fractional terms depending upon the other multiple factors such as [1 — ««]* The functions Qx are to be considered as functions with indeterminate coefficients, the degree of each such frmction being inferior by unity to that of the corresponding denominator; and it is proper to remark that the number of the indeterminate coefficients in all the frmctions 6x together is equal to the degree of the denominator fx, dx The term (aS,)*"' rf-T^ ™*y ^ reduced to the form qx g,x [i -"^ "^ [1 - a^]*-> "^ ^^'^ the functions gx being of the same degree as Ox, and the coefficients of these functions being linearly connected with those of the function 0x, The first of the foregoing terms is the only term on the right-hand side which contains the denominator [1 — a;"]* ; hence, multiplying by this denominator and then writing [1 — a;*] = 0, we find ihx which is true when x is any root whatever of the equation [1 — a;"] = 0. Now by means of the equation [1 — a;^] = 0, j~ may be expressed in the form of a rational and integral function Ox, the degree of which is less by unity than that of [1— a^]. We have therefore Gx=^gx, an equation which is satisfied by each root of [1— a^] = 0, and which is therefore an identical equation ; gx is thus determined, and the coefficients of Ox being linear functions of those of gx, the function Ox may be considered as determined And this being so, the function fx ^-" [l-af^] 140] BESEAKCHES ON THE PARTITION OF NUMBERS. 237 will be a fraction the denominator of which does not contain any power of [1 — af^] higher than [1 — «*]*"* ; and therefore 0iX can be found in the same way as 0x, and similarly OtX, and so on. And the fractional parts being determined, the integral part may be found by subtracting from ^ the sum of the fractional parts, so that the fraction J- can by a direct process be decomposed in the above-mentioned form. Particular terms in the decomposition of certain fractions may be obtained with great facility. Thus m being a prime number, assume 1 St ^ ^^ = &c. -f then observing that (1 — aJ^) = (l — ip)[l — a?^], we have for [1— a?^] = 0, 5a; = Now u being any quantity whatever and x being a root of [1 — x^"] = 0, we have identically [1 - u'**] =(t4 - x)(u-a?) ... (m - 3^^) ; and therefore putting ti = l, we have m = (l— a?) (1 — «•)... (1— a;*^*), and therefore 5a;=-, m whence 1 _o 1 1 Again, m being as before a prime number, assume = &c. 4- {l-x){\-a?)...{l-af^) ' [l-a;"']' we have in this case for [1 — af^] = 0, 0x^ which is immediately reduced to 5a? = — , . Now '' ml—x tt — a? u — x ^ ^ or putting u » 1, = (TO-l)+(m-2)a;...+a;«-»; l-x 238 RESEARCHES ON THE PARTITION OF NUMBERS. [l40 aud substituting this in the value of Oxy we find 1 « J_ (m - 1) + (tw - 2)a? ... + af^ (l-a:)(l-a;«)...(l-a?'~)'" ■*"m« [l-aJ~] The preceding decomposition of the fraction ?- gives very readily the expansion ot the fraction in ascending powers of x. For, consider a fiuction such as 0x where the degree of the numerator is in general less by unity than that of the denominator ; we have l-a;« = [l-a:«]n[l-a:«'], . where a' denotes any divisor of a (including unity, but not including the number a itself). The fraction may therefore be written under the form 0xTl [1 - Qf"'] where the degree of the numerator is in general less by unity than that of the denominator, Le. is equal to a — 1. Suppose that b is any divisor of a (including unity, but not including the number a itself), then 1 — a:* is a divisor of H [1 — sfi^, and therefore of the numerator of the fraction. Hence representing this numerator by A^ + A^x ... + Aa-iOf^', and putting a = 5c, we have (corresponding to the case 6 = 1) ^0 "H "^i + A^ ... + A^^i ^ 0, and generally for the divisor 6, Ai + Aif^i . . . + -4 (e_n6+i = 0, * ^6—1 + <4s6~i . . . + -^efr-i = 0. Suppose now that a^ denotes a circulating element to the period a, Le. write a, = 1, 9 = (mod. a), a, = in everj' other case; a frmction such as A^ + AxQ^\ . . . + Af^ia^^^^i will bo a circulating ftuiction, or circulator to the period a, and may be represented by the notation (A^y ill, •••'^•^ circlor o^. 140] RESEARCHES ON THE PARTITION 01^ NUMBERS. 239 In the case however where the coefficients A satisfy, for each divisor b of the number a, the above-mentioned equations, the circulating function is what I call a prime circulator, and I represent it by the notation (^0, ^i, ,,, ^(g^i) per Ciq. By means of this notation we have at once 0x coefficient afl in tt- — -^1= (-^o, Ai...Aa^i) per aq, and thence also 0x coefficient a?^ in (aidxY n — ;;ii ~ 5''^('^o» -^i-^-^a-i) P^r aq. Hence assuming that in the fraction ^ the degree of the numerator is less than that of the denominator (so that there is not any integral part), we have coefficient a^ in ^ = X q^(Ao, A^, ..,Aa^i) per a^; or, if we wish to put in evidence the non-circulating part arising from the divisor a = 1, coefficient .^^ in ^ = A^-^ + B^ ...^-Lq^-M + 2 /(A, A^...Aa-^) per a,; where k denotes the number of the factors 1 — a?"* in the denominator fx, a is any divisor (unity excluded) of one or more of the indices m; and for each value of- a r extends from r = to r = & — 1, where k denotes the number of indices m of which a is a divisor. The particular results previously obtained show, that m being a prime number, coefficient a^ in (i ,^)(i ^^) ^,, (i ,^^ = &c.4-^( 1,-1, 0, 0, ...) F^ ^g> and coefficient ^ in (i _^)(i _^)...(t _^„) = &c. + ^,(m-l,-l,-l. ...) perm,. Suppose, as before, that the degree of ^ is less than that of fx, and let the analytical expression above obtained for the coefficient of a;? in the expansion in ^)X ascending powers of x of the fraction ^ be represented by Fq, it is very remarkable (hx that if we expand -^ in descending powers of a?, then the coefficient of afl in this new expansion {q is here of course negative, since the expansion contains only negative powers of x) is precisely equal to —Fq; this is in fact at once seen to be 240 RESEARCHES ON THE PARTITION OF NUMBERS. [140 the case with respect to each of the partial fractions into which ^ has been de- composed, and it is consequently the case with respect to the fraction itself \ This gives rise to a result of some importance. Suppose that ^ and fx are respectively of the degrees J!V and 2); it is clear from the form of ^a? that we have /(-) = (— /a?"^; and I suppose that i>x is also such that ^ (-) =(±)'ic"'^^; then writing D^N^h, and supposing that ^ is expanded in descending powers of x, so that the coefficient of ^ in the expansion is ^Fq, it is in the first place clear that the expansion will commence with the term ar~*, and we must therefore have Fq^O for all values of q from } = — 1 to 9 = — (A — 1). Consider next the coefficient of a term ic~*~«, where 9 is or positive; the coefficient in question, the value of which is — ^(— A — 5), is obviously equal to the coefficient of a^+« in the expansion in ascending powers of x of — ~, Le. to (±)V/ coefficient a^+« in ^, . fx or what is the same thing, to (±y(-y coefficient 0^ in ^; jx and we have therefore, q being zero or positive, F(-h-q) = -(±n-yFq. In particular, when ^ = 1, Fq^O for all values of q from j = — 1 to 5 = — (D — 1) ; and q being or positive, F(^D^q)^{^r'Fq. The preceding investigations show the general form of the ftmction P(a, 6, c,...)?, viz. that P(a, b, c,...)3 = -4g*~^ + 5g'*-»...+Z5 + -If + 29^(^10, Ai,...Ai^i) per Ig, a formula in which k denotes the number of the elements a, b, c, ...^c., and I is any divisor (unity excluded) of one or more of these elements; the summation in the case of each such divisor extends from r = to r = A? — 1, where k is the number of the elements a, b, c, ...&c. of which Z is a divisor; and the investigations indicate ^ The property is a fondamental one in the general theory of deyelopments. 140] RESEARCHES ON THE PARTITION OF NUMBERS. 241 how the values of the coefficients A of the prime circulators are to be obtained. It has been moreover in eflfect shown, that ifi) = a + 6 + c + ..., then, writing for shortness P(q) instead of P(a, b, c, ...)?> ^^ have P(q) = for all values of q from } = — 1 to 5^ = — (D — 1), and that q being or positive, P(-D-q) = i-r'P(q); these last theorems are however uninterpretable in the theory of partitions, and the}' apply only to the analytical expression for P(q). I have calculated the following particular results: — P(l. 2)y =i{2? + 3 + (1, -1) per 2,1 P(l, 2, 3)9 =i^J6g» + 36? + 47 72 (. + 9(1, -1) per 2g + 8(2, -1, -1) per 3jl P(l, 2, 3. 4)9 =^|29» + 309' + 1353 + 175 + (9? + 45)(l, -1) per 2, + 32 (1, 0, - 1) per 3, + 36 (1, 0, -1, 0) per 4,1 P(l, 2, 3, 4, 5) g = gg|^ JSO g« + 900 g* + 9300 5' + 38250 g + 50651 + (13503 + 10126) (1, -1) per 2, + 3200 (2, - 1, - 1) per 3, + 5400 (1, 1, - 1, - 1) per 4, + 3466 (4, -1,-1, - 1, - 1) per 5, P(2)q =\[l + (1, - 1) per 2,| ^(2.3)9 =^{29 + 5 + 3 (1, - 1) per 2, + 4(1, -1,0) per 3, 1 C. II. 31 242 RESEARCHES ON THE PARTITION OF NUMBERS. [140 !"{% 3, 4)g 288 P(2, 3,4. 5)5 = P(2, 3, 4, 5, 6)3 = 6 ^r' + 54 5 + 107 + (18g + 81)(l, -1) per 2, + 32 (2, - 1, - 1) per 8, + 36 (1, -1, -1,1) per 4,1 + (45} + 315)(l, -1) per 2, + 160 (1, - 1. 0) per 3, + 180 (1, 0, - 1, 0) per 4, + 288 (1, - 1, 0, 0, 0) per 3,1 [lO 3« + 4009» + 5550 3»+ 31000 q + 56877 + (450 3»+ 9000? + 39075) (1, -1) per 2, (1, - 1. 0) per 3, (21, - 19, - 2) per 3, (1, 0, - 1, 0) per 4, (4, - 1, - 1, - 1, - 1) per 5, (1, -1,-2,-1, 1. 2) per 6,1 172800 + 3200 9 + 1600 + 10800 + 6912 + 4800 Pil, 2, 3, o)q = JL_ ( 720 P(\, 2, 2, 3, 4)9 = 4y' + 669"+ 3249 + 451 + 45 (1, - 1) per 2, + 80 (1, - 1, 0) per 3, + 144(1, 0, 0, 0, -1) per 5,1 g^ je 9* + 144 9" + 1194 3» + 3960 } + 4267 + (64 9" + 648 9 + 1701) (1, -1) per 2, + 256 (2, -1,-1) per 3, + 432 (1, 0, -1, 0) per 4,1 P(8)9 ^> + 1 (1, -1) per 2, + 2 (1, 0, - 1, 0) per 4, + 8(1, 0, 0, 0, -1, 0, 0, 0) per 8,1 140] RESEABCHES ON THE PARTITION OF NUMBERS. 243 + 7 (1, - 1) per 2, + 14 (1. -1,-1, 1) per 4, which are. I think, worth preserving. + 16 (3, 2, 1, 0, -1,-2,-3) per 7^ + 66 (0, -1,-1, 0, 0, 1, 1, 0) per sj, Received April 14,— Read May 3 and 10, 1855. I proceed to discuss the following problem: "To find in how many ways a number q can be made up as a sum of m terms with the elements 0, 1, 2, . . . A;, each element being repeatable an indefinite number of timea" The required number of partitions is represented by P(0, 1, %...k)^q, and the number of partitions of q less the number of partitions of j — 1 is repre- sented by F(0, 1, 2, ...&)«}. We have, as is well known, P(0. 1. 2....A)»3 = coeffieient ^^ in (i _,)(i _i)...(i _^,) . where the expansion is to be effected in ascending powers of z. Now 1 -i.lz^' (l-a^-^0(l-a^+«) , the general term being (1 - a^+0(l - a:*+») ... (1 - a^^"*) (l-a:)(l-a;»)... {l-af^) or, what is the same thing, (l-a:)(l-a;«)... (1-a?*) and consequently P(0. 1. 2. ...*)-.5 = eoeflBeient ^ in ^' (i-l^d -C)..r(i -S) ' to transform this expression I make use of the equation (l+a:^)(l+.;»^)...(l+.*^) = l+-L__^+ (\,^/(\^^) V + &C., 31—2 z^, s^. 244 RESEARCHES ON THE PARTITION OF NUMBERS. [140 where the general term is and the series is a finite one, the last term being that corresponding to s^k, viz. ^(i+i)^ Writing —a^ for z, and substituting the resulting value of (I - a-H^i) (1 - -r~+«) . . . (1 - a-»+*) in the formula for P(0, 1, 2, ...4)"y, we have P(0. 1. 2. ... A)-v = S,^(-y coeffideot ^ in (i.^)(i_^)...(i_^)(l-^)(l-<'^)-.(l-x*n ' where the summation extends firom ^^0 to s^k; but if for any value of s between these limits «m + ^(«+l) becomes greater than q, then it is clear that the summation need only be extended firom 8=^0 to the last preceding value of s, or what is the same thing, firom « = to the greatest value of s for which j — «m — ^«(«+l) is positive or zenx It is obvious, that if y > km, then P(0, 1, 2...A)-}«0; and moreover, that if ^ > Ikm, then P(0, 1, 2,...it)~^«P(0, 1, 2,...it)~.itm-tf, so that we may always suppose j > Jibm. I write therefore q = ^(kni'-a) where a is xero or a {positive integer not greater than km, and is even or odd according as Arm 18 oven or inld. Substituting this value of q and making a slight change in the form of the n^sult, we have whon« tht^ Hummation extends firom 9 = to the greatest value of s for which (U* -«)'H - ^ot- i«(^+ 1) IB positive or zero. But we may, if we please, consider the miiuiimtion m extending, when k is even, fix)m 8^0 to 8^\k — \, and when k is odd, \\\\\\\ /i-O to j» "• i (il-' 1); the terms corresponding to values of 8 greater than the gh»Hti»Hi value for which (iifc-«) m- Ja- J«(«4- 1) is positive or zero being of course equal to X(«rt»« It may be noticed, that the firaction will be a proper one if ri* (k H)(k-H'k-\)\ or substituting for 8 its greatest value, the firaction will be a iiiniirr Olio for all values of «, if, when it is even, o< Jifc(A: + 2), and when k is odd, «• \(k\ \)(k \ HV Wo have in a nimilar maimer, /••((). I. 2. ..Arv = coefficient «•,« in (-r -^)(l-^X(l -^*^) ' 140] RESEABCHE8 ON THE PARTITION OF N(JMB£R8. 245 which leads to ^(0, 1, 2...ik)~i(Jtm-a) = where the summation extends, as in the former case, firom « = to the greatest value of «, for which (Jifc — «)m — ^a — i«(« + 1) is positive or zero, or, if we please, when k is even, from « = to « = iA?— 1, and when 8 is odd, from « = to « = J(A — 1). The condition, in order that the fruction may be a proper one for all values of s, is, when k is even, a + l<\k(k + 2), and when k is odd, a + 1 < i(A + l)(A: + 3). To transform the preceding expressions, I write when k is odd jc^ instead of x, and I put for shortness instead of ^k — s or 2(iA: — «), and y instead of Ja-h J«(«-»-l) or a + « (« + 1) ; we have to consider an expression of the form coefficient a^ in =r , where Fx is the product of factors of the form 1 — «:*. Suppose that a is the least common multiple of a and 0, then (1 — a*') -h (1 — «*) is an integral function of x, equal x^ suppose, and 1 -?- (1 — a;*) = ^^ -s- (1 - a?*'). Making this change in all the factors of Fx which require it (Le. in all the factors except those in which a is a multiple of 0), the general term becomes coefficient ar^ in —tz — , where Go; is a product of factors of the form 1 — af^'y in which a' is a multiple of 0, i.e. Go; is a rational and integral fiinction of of. But in the numerator a!*Hx we may reject, as not contributing to the formation of the coefficient of a^, all the terms in which the indices are not multiples of 0\ the numerator is thus reduced to a rational and integral function of afy and the general term is therefore of the form coefficient af^ in -Wv* or what is the same thing, of the form coefficient af* m — , icx where xx is the product of factors of the form 1 — ic*, and Xo; is a rational and integral function of x. The particular value of the fraction depends on the value of s: and uniting the dififerent terms, we have an expression \x coefficient x^ in 8m (—V — , ^ KX ^hich is equivalent to coefficient a"* in ^ , fx 246 RESEARCHES ON THE PARTITION OF NUMBERS. [l40 where /i; is a product of factors of the form 1 ---af^, and ^ is a rational and integral function of x. And it is clear that the fraction will be a proper one when each of the fractions in the original expression is a proper Auction, i.e. in the case of P(0, 1, 2...Jk)'»J(*^-«)» when for k even, a<iik(A:+2), and for k odd, a<i(Jfc+l)(it+3); and in the case of P'{0, 1, 2 ... A:)^ J(A?m — a), when for k even, a+l<\k(k + 2)y and for k odd, a+1 <l{k-\-l){k-hS). We see, therefore, that P(0, 1, 2...ifc)«»K*wi-aX and 1^(0, 1, 2...ifc)«i(^-«X are each of them of the form coefficient af* in ?-, where fx is the product of factors of the form 1 — a*, and up to certain limiting values (dX of a the fiuction is a proper fraction. When the fraction ~- is known, we may there- fore obtain by the method employed in the former part of this Memoir, anal3rtical expressions (involving prime circulators) for the functions P and P', As an example, take P(0, 1, 2, 3)'^|m, which is equal to 1 coefficient a^ in — coefficient af* in (l-a;«)(l-a?*)(l-««) (l-aj«)(l-ic»)(l -«?•)• The multiplier for the first fraction is which is equal to 1+ a;* -h ar* + ic« + 2a;» + a^' + «*'. Hence, rejecting in the numerator the terms the indices of which are not divisible by 3, the first term becomes coefficient ac^ in (l-a^)(l-aj»«)(l-a^)' or what is the same thing, the first term is 1+a^ + a?* coefficient xl^ in (l-a^)»(l-a?*)' 140] RESEARCHES ON THE PARTITION OF NUidBERS. 247 and, the second term being — coefficient aS^ in (l-a:»)«(l-a?*)' 1 +ip* we have P(0, 1, 2, 3)'"|w = coeflBcient a^ in /i ^a^vn --ar<V and similarly it may be shown, that P(0, 1, 2, 3)"* i(3w - 1) = coefficient af» in (l-ic«)»(l-ar*) As another example, take P'(0, 1, 2, 3, 4, 5)fm, which is equal to 1 coefficient a*^ in — coefficient a^ in + coefficient off^ in (1 - a?*) (1 - a^) (1 - ic*) (1 - a?io) (^ (1 - a;«) (1 - «?•) (1 - a^) (1 - a:«) (1 - a;«) (1 - a?*) (1 - a;*) (1 - ««) • The multiplier for the first firaction is which is a function of a? of the order 36, the coefficients of which are 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 4, 4, 6, 4, 6, 5, 7, 5, 7, 5, 7, 5, 6, 4, 6, 4, 4, 3, 4. 2, 3, 1, 2, 1, 1, 0, 1, and the first part becomes therefore coefficient aS^ m — -p. ^r-r- — --r—- — —^—- — — — . (1 -a^)(l -a^)(l -aj'Xl -aJ®) The multiplier for the second fraction is (l-a^)(l-a^')(l-a;»*) (l-a;»)(l-aj*)(l -a^)' which is a function of a? of the order 14, the coefficients of which are 1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1 ; and the second term becomes iX3- -L «• 2a^ + 2a:* + 3a^ + «" + a:" -coefficient a^ m (i ^^y^x -ar^)(l -a:") > 248 RESEABCHES ON THE PARTITION OF NUMBERS. [UO and the third term is coefficient «** in Now the fractions may be reduced to a common denominator (1 -a;«)(l -a?*)(l -«•)(! -«!») 1 — a^ by multipljring the terms of the second fraction by :. — — (= 1 + a;* + a?*), and the terras 1 — «" of the third Auction by ^ ^ (= 1 + a?*) ; performing the operations and adding, the numerator and denominator of the resulting fraction will each of them contain the factor 1 — a^ ; and casting this out, we find P(0, 1, 2, 3, 4, o)** fm = coefficient ai^ in (l-a?*)(l-a;«)(l-a:»)' I have calculated by this method several other particular cases, which are given in my "Second Memoir upon Quantics", [141], the present researches were in fact made for the sake of their application to that theory. Received April 20,— Read May 3 and 10, 1855. Since the preceding portions of the present Memoir were written, Mr Sylvester has communicated to me a remarkable theorem which has led me to the following additional investigations ^ Let ^ be a rational fraction, and let (a? — a?i)* be a &ctor of the denominator fr, /« then if denote the portion which is made up of the simple fractions having powers of x — Xi for their denominators, we have by a known theorem . ^l = coefficient - in %j — : — ( . Now by a theorem of Jacobi's and Cauchy's, coefficient - in ^f: = coefficient - in F(yyt)'^'t] Z V whence, writing Xi + z=^ ^«~S we have J^l = coefficient - in -i ^; ^ . (/a:],, t x^-x^f(xi€r*) 1 Mr Sylvester's researohes are published in the Quarterly Mathematical Journal^ July 1855, [vol. i. pp. 141—152], and he has there given the general formula as well for the eiroulating as the non-dronlating part of the expression for the number of partitions.— Added 28rd February, 1856.— A. G. 140] RESEARCHES ON THE PARTITION OF NUMBERS. 249 Now putting for a moment x=^x^(f, we have 1 1 1^1 + ^tf — 77 T\ + • • • » x^-x^ a:i(l-6'+*) iri(l-6') ' "^ x^{\-^) and 3^ = ar9jr, whence Xi^xtf Xi — x 1 Xi — x 1.2^ Xi — x the general term of which is fS-l Hence representing the general term of Xi<f> (XifT*) by x^i<~*, 80 that X^i = coefficient ^ m ^' f(x.e-*) ' we find, writing down only the general term, i/4*r *'* "^ n(*- 1) ("^^^U-^"^ ••• ' where the value of x^ depends upon that of «, and where 8 extends from « = 1 to « = Ar. Suppose now that the denominator is made up of £sictors (the same or different) of the form 1 — a?"*. And let a be any divisor of one or more of the indices m, and let k be the number of the indices of which a is a divisor. The denominator contains the divisor [1 — af^f, and consequently if p be any root of the equation [!—«"] = 0, the denominator contains the fetctor (p — a?)*. Hence writing p for Xi and taking the sum with respect to all the roots of the equation [1 — ic*] = 0, we find vhere vp = coefficient - in ^* jv^^v > and as before 8 extends from ^ = 1 to 8==k. We have thus the actual value of the function $x made use of in the memoir. A preceding formula gives coefficient t in S^\ f ^Hich is a very simple expression for the non-circulating part of the fraction ^ ^*^is is, in fact, Mr Sylvester's theorem above referred to. c. n. 32 250 [141 141. A SECOND MEMOIR UPON QUANTICS. [From the Philosophical Transactiona of the Royal Society of London, vol. CXLVI. for the year, 1856, pp. 101—126. Received April 14,— Read May 24, 1855.] The present memoir is intended as a continuation of my Introductory Memoir upon Quantics, t. CXLIV. (1854), p. 245, and must be read in connexion with it ; the paragraphs of the two Memoirs are numbered continuously. The special subject of the present memoir is the theorem referred to in the Postscript to the Introductory Memoir, and the various developments arising thereout in relation to the number and form of the covariants of a binary quantic. 25. I have already spoken of asyzygetic covariants and invariants, and I shall have occasion to speak of irreducible covariants and invariants. Considering in general a function u determined like a covariant or invariant by means of a system of partial differential equations, it will be convenient to explain what is meant by an asyzygetic integral and by an irreducible integral. Attending for greater simplicity only to a single set (a, 6, c, . . .), which in the case of the covariants or invariants of a single function will be as before the coefficients or elements of the function, it is assumed that the system admits of integrals of the form u^ P, u ^ Q, &c., or as we may express it, of integrals P, Q, &c., where P, Q, &c. are rational and integral homogeneous functions of the set (a, 6, c, ...), and moreover that the system is such that P, Q, &c. being integrals, <f>(P, Q, •*•) is also an integral Then considering only the integrals which are rational and integral homogeneous functions of the set (a, b, c, ...), integrals P, Q, JB, ... not connected by any linear equation or syzygy (such as \P + fiQ + vR ... 0),Q) are said to be asyzygetic; but in speaking of the asyzygetic integrals of a particular degree, it is implied that the integrals are a system such that every other integral of 1 It is hardly necessary to remark, that the multipliers X, fi, r, ... , and generally any eoefficients or quantities not expressly stated to contain the set (a, b, c, ...), are considered as independent of the set, or to use a conyenient word, are considered as "trivials." 141] A SECOND MEMOIR UPON QU ANTICS. 251 the same degree can be expressed as a linear function (such as XP + /tQ + ]/i2...) of these integrals; and any integral P not expressible as a rational and integral homo- geneous fnnction of integrals of inferior degrees is said to be an irreducible integral 26. Suppose now that A^^ A^, A^, &c. denote the number of asyzygetic integrals of the degrees 1, 2, 3, &c. respectively, and let a^, a,, a,, &c. be determined by the equations A = iai(ffi + l) + or„ -^8 = i «! («! + l)(ai + 2) + aia,-hor3, -44 = ^ai(ai+l)(«i+2)(ai + 3)-|-iai(ori-hl)a>-l-aia, + ia,(a,+ l)-ha4, &c., or what is the same thing, suppose that . l + A^x + A^-\-&c. =(l-a:)'^*(l-a;«)"^(l-a:»)"^...; a little consideration will show that a^, represents the number of irreducible integrals of the degree r less the number of linear relations or syzygies between the composite or non-irreducible integrals of the same degree. In £sict the asyzygetic integrals of the degree 1 are necessarily irreducible, i.e. Ai^a^. Represent for a moment the irreducible integrals of the degree 1 by X, X\ &c., then the composite integrals Z*, XX\ &c., the number of which is iai(ai + l), must be included among the asyzygetic integrals of the degree 2; and if the composite integrals in question were asyzygetic, there would remain ilj — i «!(«! + 1) for the number of irreducible integrals of the degree 2 ; but if there exist syzygies between the composite integrals in question, the number to be subtracted from A^ will be ^^^(ai + l) less the number of these syzygies, and we shall have -^a — i ai(ai + l), ie. ffj equal to the number of the irreducible integrals of the degree 2 less the number of syzygies between the composite integrals of the same degree. Again, suppose that Oj is negative = — A, we may for simplicity suppose that there are no irreducible integrals of the degree 2, but that the com- posite integrals of this degree, X^, XX', &c., are connected by fi^ syzygies, such as \X* + fiXX' + &C. = 0, XiZ* + fiiXX' + &c. = 0. The asyzygetic integrals of the degree 4 include X\ X*X', Sec, the number of which is ^ Oi (ce, + 1) (Oi + 2) (a^ + 3) ; but these composite integrals are not asyzygetic, they are connected by syzygies which are angmentatives of the fi^ syzygies of the second degree, viz. by syzygies such as (XX« + fiZZ'...)Z» = 0, (XZ»-h/iZZ'...)ZZ' = 0, &c. (XiZ«+/AiZZ'...)Z« = 0, (XiZ» + /AiZZ'...)ZZ' = 0, &c., the number of which is iai(ai-hl))8j. And these syzygies are themselves not asyzygetic, they are connected by secondary syzygies such as Xj(VZ« + m-X'Z'...)Z» + /Ii(XZ« + /iaZZ'...)ZZ' + &c. -X(XiZ>+/iiZZ'...)Z»-fi(XiZ« + AhZZ'...)ZZ'-&c. = 0, &c. &c., 32—2 252 A SECOND MEMOm UPON QUANTIGS. [141 the number of which is i)9s09, — 1). The real number of syzygies between the com- posite integrals X\ X*X\ &c. (Le. of the syzygies arising out of the Pt syzygies between X*, XX\ &c.), is therefore i ai(ai + 1)A — ii8,()8j— 1), and the number of integrals of the degree 4. arising out of the integrals and syzygies of the degrees 1 and 2 respectively, is therefore or writing —a, instead of )8,, the number in question is A «!(«! + l)(or, + 2)(ai -h 3) + i ai(a, + 1) or, + i a, (a, + 1). The integrals of the degrees 1 and 3 give rise to ttiO, integrals of the degree 4; and if all the composite integnJs obtained as above were asjrzygetic, we should have ^4-Affi(ai + l)(ai + 2)(aj + 3)-iai(ar,-hl)a,--ia,(a,-hl)-aia,, i.e. OL^ as the number of irreducible integrals of the degree 4; but if there exist any further syzygies between the composite integrals, then a^ will be the number of the irreducible integrals of the degree 4 less the number of such further syzygies, and the like reasoning is in all cases applicable. 27. It may be remarked, that for any given partial differential equation, or system of such equations, there will be always a finite number v such that given v independent integrals every other integral is a function (in general an irrational fimction only expressible as the root of an equation) of the v independent integrals; and if to these integrals we join a single other integral not a rational function of the v integrals, it is easy to see that every other integral will be a rational function of the y + 1 integrals : but every such other integral will not in general be a rational and integral fimction of the ]/ + 1 integrals ; and lincorrecf] there is not in general any finite number whatever of integrals, such that every other integral is a rational and integral function of these integrals, Le. the number of irreducible integrals is in general infinite ; and it would seem that this is in fact the case in the theory of covarianta 28. In the case of the co variants, or the invariants of a binary quantic, A^ \a given (this will appear in the sequel) as the coefficient of o;^ in the development, in ascending powers of a?, of a rational fi:uction ^ , where yi is of the form (1 - a:)^(l - «•)^ . .(1 - a:*y», and the degree of ^ is less than that of ^. We have therefore and consequently <^=(1 -a?)*-Xl-aj«y*-"-...(l -af'Y'^O -a*+')"**^'- • 141] A SECOND MEMOIR UPON QU ANTICS. 253 Now every rational factor of a binomial l—af^iB the irreducible factor of 1 — af^', where m is equal to or a submultiple of m. Hence in order that the series a^, or,, 3s»*-- may terminate, ^ must be made up of factors each of which is the irreducible factor of a binomial 1 — af^, or if ^ be itself irreducible, then <f>x must be the irreducible factor of a binomial 1 — a^. Conversely, if ^ be not of the form in question, the series a^, a,, a^, &c. will go on ad infinitum, and it is easy to see that there is no point in the series such that the terms beyond that point are all of them negative, i.e. there will be irreducible covariants or invariants of indefinitely high degrees; and the number of covariants or invariants will be infinite. The number . of invariants is first infinite in the case of a quantic of the seventh order, or septimic ; the number of covariants is first infinite in the case of a quantic of the fifth order, or quintic. [As is now well known, these conclusions are incorrect, the number of irreducible covariants or invariants is in every case finite.] '2d. Resuming the theory of binary quantics, I consider the quantic (a, 6,...6\ a^$a?, y)*". Here writing [ady] = mbda + (m — 1 ) cS^. . .+ a'9y, = )', any function which is reduced to zero by each of the operations X — yd^, Y—xdy is a CO variant of the quantic. But a co variant will always be considered as a rational and integral function separately homogeneous in regard to the facients {x, y) and to the coeflBcients (a, 6,...6\ d). And the words order and degree will be taken to refer to the facients and to the coeflBcients respectively. I commence by proving the theorem enunciated. No. 23. It follows at once from the definition, that the covariant is reduced to zero by the operation X — ydx . Y — xdy — Y— xdy . X — yd^t which is equivalent to X . F— Y ,X+ydy — aSx' Now X.Y = XY+X(Y) Y.X=YX+Y(X\ where XY and YX are equivalent operations, and X(F)= lmada-^2(m-l)bdt...+ mlb'dt', Y(X)= wl6aft...+ 2(m- ])b'dt-^\„ia:da, whence Jf (F)- F(Jf) = 7yia9a + (m — 2)63ft...-(m-2)6'9e> -ma'3a, =* suppose, and the covariant is therefore reduced to zero by the operation k + ydy — a^x- Now as regards a term a*b^..M^d''\a^y^, we have fc = wa + (i?i-2)/8..., '-(m-2)/3"-md ydy-aSx^'j-i; 254 A SECOND MEMOIR UPON QUANTIC8. [141 and we see at once that for each term of the covariant we must have ma + (m - 2) /8...- (m - 2) ;8^ - tna' + j - 1 = 0, i.e. if (x, y) are considered as being of the weights ^, —^ respectively, and (a, b,...b\ a) as being of the weights — |fn, — ^m + 1, ...^m— 1, ^m respectively, then the weight of each term of the covariant is zero. But if {x, y) are considered as being of the weights 1, respectively, and (a, 6,...6\ a) as being of the weights 0» l,...m — 1, m respectively, then writing the equation under the form and supposing that the covariant is of the order ^ and of the degree 0, each term of the covariant will be of the weight \ (mO + /i). I shall in the sequel consider the weight as reckoned in the last-mentioned manner. It is convenient to remark, that as regards any function of the coefficients of the degree 6 and of the weight 9, we have X.Y-Y.X^md-iq. 30. Consider now a covariant {A, B,...B, A^\x, yY of the order n and of the degree 0\ the covariant is reduced to zero by each of the operations X—yd,, T — aldy, and we are thus led to the systems of equations XA^O, XB = ijlA, XC = (jjL-l)B. Xff= 2(7. XA'=ff ; and YA = B, YB = 2(7, Y(T= 0* - 1) F, Yff= ,iA\ YA'= 0. Conversely if these equations are satisfied the function will be a covariant. I assume that il is a function of the degree and of the weight ^ (md — /i), satisfying the condition 141] A SECX)ND MEMOm UPON QUANTICS. 255 and I represent by YA, FM, Y^A, &c. the results obtained by successive operations with Y upon the function A, The function Y'A will be of the degree and of the weight ^(mO -- fi) + 8, And it is clear that in the series of terms YA, FM, FM, &e., we must at last come to a term which is equal to zero. In fact, since m is the greatest weight of any coeflBcient, the weight of F* is at most equal to m0, and therefore if i(md — /i)+«>m5, or 8 > ^(mO -^^ fi), we must have Y'A=0. Now writing for greater simplicity XY instead o{ X,Y, and so in similar cases, we have, as regards Y'A, XY-YX = fi-28. Hence ^od consequently i*-5iinilarly ^ therefore d again, therefore ^nerally {XY-'YX)A = fiA, XYA=^YXA'\-lj^A=ij^A, (ZF- FX) F^ = (/i - 2) Fil, X Y^A = FZ Fil + (/i - 2 ) Fil = /aF4 + (a*- 2) F.4 = 2(/i- l)Fil. (XF-FZ)F«il=(/A-4)FM, XY^A = YXY^A + (/i - 4) Y^A = 2(/t- 1) F«il + 0i- 4)FM = 3 (/i- 2) FM, ZF'^=«(/A-« + l)FM. ^ce putting « = ^+l, /i + 2, &c., we have ZF'*+M = 0, XYi'^^A = - (/i + 2) 1 . F'*+'^, ZF'*+M=-(/i + 3)2.F'*-«J, &c, F^+M = ; y^*^ unless this be so, La if F''+*-4 + 0, then from the second equation Z F'^+M + 0, and ^^^fore Yf'^A^O, from the third equation ZF'*+»=NO, and therefore Z'^+M^O, and so ^ dd infinitum^ Le. we must have Y'^'^^A = 0. ^^^^^ations which show that 256 A SECOND MEMOIR UPON QUANTIC8. [141 81. The Buppoeitiona which have been made as to the function A, give therefore the equations XA =0. XYA =,iA. XY*A = 2(ji-\)YA, XYi^A'=(tYi'*'A, Yi'+'A = ; and if we now assume the sratem becomes XA'=0, XB = fiA, XC = (ji-l)B. J A' = F, YA' = 0; 80 that the entire system of equations which express that (A, B.,.B, ^^^^f vY ^ a covariant is satisfied ; hence Theorem. Given a quantic (a, 6, ...6\ a'$a?, y)***; if il be a function of the coefficients of the degree and of the weight ^ (mO — fi) satisfying the condition XA =0, and if -B, C, ... jB", -4' are determined by the equations B^YA, C = ^YB,,..A' = ^YB\ then will (A, B,...B, A\x, yY be a covariant. In particular, a function A of the degree and of the weight ^m^, satisfying the condition XA = 0, will (also satisfy the equation YA = and vdll) be an invariant. 32. I take now for A the most general function of the coefficients, of the degree 6 and of the weight \ {mO — /x) ; then XA is a function of the degree 6 and of the weight ^(m^ — /i)— 1, and the arbitrary coefficients in the function A are to be determined so that XA =■ 0. The number of arbitrary coefficients is equal to the number of terms in A^ and the number of the equations to be satisfied is equal to the number of terms in XA ; hence the number of the arbitrary coefficients which remains indeter- minate is equal to the number of terms in A less the number of terms in XA ; and since the covariant is completely determined when the leading coefficient is known, 141] A SECOND MEMOIR UPON QUANTICS. 257 the diflference in question is equal to the number of the asyzygetic covariants, i.e. the number of the asyzygetic covariants of the order fi and the degree is equal to the number of terms of the degree and weight \ (mO — fi), less the number of terms of the degree d and weight J(md— /a)— 1. 83. I shall now give some instances of the calculation of covariants by the method just explained. It is very convenient for this purpose to commence by forming the literal parts by Arbogast's Method of Derivations : we thus form tables such as the following :— a b c a* ah ac 6« be 6» a b c ' ! ah 6« ad be bd ed d" ! rt* a^b a*c aH abd aed cuP bd" ed" (P ah^ ahe b" ac" b^e b^d be" bed c'd a* M a»<J aH a^bd a^ed a^d" abcP a^<P ad^ bd^ ccP d* a«6» a^be a^c" ab'd abed ac'd b^d" bcd^ c^rf* ab* ah^c 6* abc" b^c bH 6V b^ed be" bc'd eH a b c d e a» ab ac ad M be bd ed d' V he bd ed <? C. II. 33 258 A SECOND MEMOIB UPON QUANTICS. [141 a' a*6 a^c a^d a«« abe ace ode oe* W ce» de* ^ ah* abc abd acd cuP bee bde cde d^. &» b*d be" bh bed c'd d^ - <? 34. Thus in the case of a cubic (a, 6, c, d$a?, y)*, the tables show that there will be a single invariant of the degree 4. Represent this by •\' Babcd 4- Cad" + Dl^d which is to be operated upon with a36 + 263c + 3c9d- This gives Le. E^ + B + 6i4 a^cd + 3/> + 25 ab\l + '2E + 6C + 45 putting . + 35 + 35 abc* b'c 5 = 0, &c.; or zl = 1, we fi nd 5 = -(], C=4, Z> = 4. — 3, and the invariant is — 6a6cd + 4ac» + 46»d -36V. Again, there is a covariant of the order 3 and the degree 3. The coefficient of .r* leading coefficient is Aa*d •i- Babe which operated upon with a3e> + 2ft3c4- 3c9d, gives + 5 + 3C 1 + 2Z? + 3i4 a'e ab' 141] A SECOND MEMOIR UPON QUANTICS. 259 i.e. B-^SA =0, 3C+aB = 0; or putting ^ = 1, we have 5 = -3, C = 2, and the leading coefficient is - Sabc + 2 6». The coefficient of ah/ is found by operating upon this with (369a + 2cdi, + ddc), this gives abd ac' b'c + 6 -9 -6 + 12 -3 i.e. the required coefficient of a^y is Sabd - 6ac» 4-3 6»c; and by operating upon this with J (369a + ^cdb + dde\ we have for the coefficient of a?y^ acd b'd be' 4. » + If -9 + 3 + 6 -6 4. » i.e. the coefficient of xi/* is — 3 acd + 66»d ~3 6c». Finally, operating upon this with ^ (369a + 2c9ft + d9c)> we have for the coefficient of y*, -1 -3 + 8 -2 -2 i.e. the coefficient of y* is and the covariant is + 36cd -2c», ad' bed ( abe-3 V +2 oc* -6 6*c +3 h*d +6 ftc" -3 1 6e(^ + 3 c» -2 $a^» yf [I now write the numerical coefficients after instead of before the literal terms.] 33—2 260 A SECOND MEMOIR UPON QU ANTICS. [141 I have worked out the example in detail as a specimen of the most convenient method for the actual calculation of more complicated covaiiants^ 35. The number of terms of the degree and of the weight q is obviously equal to the number of wajrs in which q can be made up as a sum of 6 terms with the elements (0, 1, 2, ...m), a number which is equal to the coeflBcient of aflg^ in the development of ___^ 1 and the number of the asyzygetic covariants of any particular degree for the quantic (♦$^> y)** can therefore be determined by means of this development. In the case of a cubic, for example, the function to be developed is {\ " z){l - xz) {\ -a^z) {I -a?zy which is equal to where the coeflScients are given by the following table ; on account of the symmetry, the series of coefficients for each power of ^r is continued only to the middle tenn or middle of the series. 1 1 1 1 1 2 2 1 1 2 3 3 1 1 2 3 4 4 5 1 I 2 3 4 5 6 6 1 1 2 3 4 5 7 7 8 8 (0) (1) (2) (3) (4) (5) (6) ^ Note added Feb. 7, 1856.— The following method for the calculation of an invariant or of the leading coefficient of a covariant, is easily seen to be identical in principle with that given in the text Write down all the terms of the weight next inferior to that of the invariant or leading coefficient, and operate on each of theee separately with the symbol ind. 6 . - + 2 ind. c . r +...(»»- 1) ind. 6* . -^ , where we are first to multiply by the fraction, rejecting negative powers, and then by the index of the proper letter in the term so obtained. Equating the results to zero, we obtain equations between the terms of the invariant or leading coefficient, and replacing in these equations each term by its numerical coefficient in the 141] A SECOND MEMOIR UPON QUANTICS. 261 and from this, by subtracting from each coefficient the coefficient which immediately precedes it, we form the table: (0) L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( } 1 1 1 1 1 2 1 (2) (3) (5) The successive lines fix the number and character of the covariants of the degrees 0, 1, 2, 3, &C. The line (0), if this were to be interpreted, would show that there is a single covariant of the degree ; this covariant is of course merely the absolute con- stant unity, and may be excluded. The line (1) shows that there is a single covariant of the degree 1, viz. a covariant of the order 3; this is the cubic itself, which I represent by U. The line (2) shows that there are two asyzygetic covariants of the degree 2, viz. one of the order 6, this is merely JJ\ and one of the order 2, this I represent by H, The line (3) shows that there are three asyzygetic covariants of the degree 3, viz. one of the order 9, this is JJ* ; one of the order 5, this is UH, and one of the order 3, this I represent by 4). The line (4) shows that there are five asyzygetic covariants of the degree 4, viz. one of the order 12, this is U^\ one of the order 8, this is V^H\ one of the order 6, this is JT*; and one of the order 0, ie. an invariant, this I represent by V. The line (5) shows that there are six asyzygetic covariants of the degree 5, viz. one of the order 15, this is IT" ; one of the order 11, this is U^H ; one of the order 9, this is I7*4> ; one of the order 7, this is UH*\ one of the order 5, this is H^ ; and one of the order 3, this is V CT. The line (6) shows that there are 8 asyzygetic covariants of the degree 6, viz. one of the order 18, this is U^\ one of the inyariant or leading coefficient, we have the equations of connexion of these nomerioal coefficients. Thns, for the dlBcriminant of a cnhic, the terms of the next inferior weight are ahd^ ab^d, abc\ &*e, and operating on each of these separately with the symhol ind. 6 . - + 2 ind. c . ■=• + 3 ind. d . - , a b c we find abed + 6a«d' 3bH + 2abed 2 6»c« + 6ac« ■\-Sabcd + 4fc»c> + 8 6»d and equating the horizontal lines to zero, and assuming a'd':=l, we have a^=l, abed= -6, ac^=4, b*d=if 6M= - 3, or the value of the discriminant is that given in the text. 262 A SECX)ND MEMOIR UPON QU AN TICS. [141 order 14, this is U*H \ one of the order 12, this is 17** ; one of the order 10, thi^ is U^H^\ one of the order 8, this is UH^\ two of the order 6 (Le. the three covariants ff', ^ and Vf/"* are not asyzygetic, but are connected by a single linear equation or syzygy), and one of the order 2, this is VH. We are thus led to the irreducible covariants U, JT, ^, V connected by a linear equation or syzygy between H*, ^' and VfT', and this is in fact the complete system of irreducible covariants; V is therefore the only invariant. 36. The asyzygetic covariants are of the form U^H^V^^ or else of the form U^H^V^^; and since U, H, V are of the degrees 1, 2, 4 respectively, and ^ is of the degree 3, the number of asyzygetic covariants of the degree m of the first form is equal to the coefficient of a^ in 1 -r- (1 — x) (1 — a^) (1 — a^), and the number of the asyzygetic covariants of the degree m of the second form is equal to the coefficient of w'^ in a^^(l— a:) (1 — ic")(l — a?*). Hence the total number of asyzygetic covariants is equal to the coefficient of »"* in (1 + a;*) -^ (1 — x) (1 — x*) (I — a^), or what is the same thing, in and conversely, if this expression for the number of the asyzygetic covariants of the degree m were established independently, it would follow that the irreducible invariants were four in number, and of the degrees 1, 2, 3, 4 respectively, but connected by an equation of the degree 6. As regards the invariants, eveiy invariant is of the form V^, i.e. the number of asyzygetic invariants of the degree m is equal to the coefficient of ic*" in , ^ , and conversely, fi-om this expression it would follow that there was a single irreducible invariant of the degree 4. 37. In the case of a quartic, the function to be developed is: 1 (1 - £r) (1 - arz) (1 - a:»z) (1 - a^^) (1 - a:*^) ' and the coefficients are given by the table. (0) 0) (2) (3) (4) (5) (6) 1 1 1 1 1 1 2 2 3 1 1 2 3 4 4 5 1 1 2 3 5 5 7 7 8 1 1 2 3 5 6 8 9 11 11 12 1 1 2 3 5 6 9 10 13 14 16 16 18 141] A SECOND MEMOIR UPON QUANTICS. 263 and subtracting from each coefficient the coefficient immediately preceding it, we have the table : 1 (0) 1 (1) 1 1 1 (2) 1 1 1 1 1 1 1 (3) 1 1 t 1 2 2 (4) 1 1 1 1 2 3 1 1 2 1 2 1 (5) 1 1 1 2 3 1 2 2 (6) the examination of which will show that we have for the quartic the following irreducible covariants, viz. the quartic itself U; an invariant of the degree 2, which I represent by / ; a covariant of the order 4 and of the degree 2, which I represent by H ; an invariant of the degree 3, which I represent by J; and a covariant of the order 6 and the degree 3, which I represent by <I> ; but that the irreducible covariants are connected by an equation of the degree 6, viz. there is a linear equation or syzygy between <!>», PH^, PJH^U, IJ^HU^ and J^U^\ this is in fact the complete system of the irreducible covariants of the quartic: the only irreducible invariants are the invariants /, J. 38. The asyzygetic covariants are of the form U^I'^H^J*, or eke of the form UpI'^H^'J*^, and the number of the asyzygetic covariants of the degree m is equal to the coefficient of a^ in (1 + ir')^(l — a:)(l — a:^)*(l — a^), or what is the same thing, in 1-A^ (i-j')(i'-a^y(i-x'y' and the asyzygetic invariants are of the form /pJ^, and the number of the asyzygetic invariants of the degree m is equal to the coefficient of af^ in 1-t-(1 — a;*)(l — a^). Conversely, if these formulae were established, the preceding results as to the form of the system of the irreducible covariants or of the irreducible invariants, would follow. 39. In the cas^ of a quintic, the function to be developed is (I " z)(l - xz){l - a^z)(l ^ .z^z) (I -a^z)(l-a^z)' and the coefficients are given by the table : 264 A SECOND MEMOIR UPON QUANTICS. [1 1 1 1 1 1 1 2 2 3 3 1 1 2 3 4 5 6 6 1 1 2 3 5 6 8 9 11 11 12 1 1 2 3 5 7 9 11 U 16 18 19 20 (0) (1) (2) (3) (4) (5) and subtracting from each coefficient the one which immediately precedes it, we h the table : 1 1 1 1 1 1 1 1 1 1 1 1 (0) 0) (2) (3) (4) (5) We thus obtain the following irreducible covariants, viz. : Of the degree 1 ; a single covariant of the order 5, this is the quintic itself Of the degree 2 ; two covariants, viz. one of the order 6, and one of the order 2 Of the degree 3 ; three covariants, viz. one of the order 9, one of the order 5, i one of the order 3. Of the degree 4 ; three covariants, viz. one of the order 6, one of the order 4, i one of the order (an invariant). Of the degree 6 ; three covariants, viz. one of the order 7, one of the order 8, i one of the order 1 (a linear covariant). These covariants are connected by a single syzygy of the degree 5 and of order 11 ; in fact, the table shows that there are only two asyzygetic covariants this degree and order; but we may, with the above-mentioned irreducible covaria of the degrees, 1, 2, 3 and 4, form three covariants of the degree 5 and the or 11 ; there is therefore a syzygy of this degree and order. 141] A SECOND MEMOIR UPON QU ANTICS. 265 40. I represent the number of ways in which q can be made up as a sum of m terms with the elements 0, 1, 2, . . . m, each element being repeatable an indefinite number of times by the notation P(0, 1, 2, ...m/j, and I write for shortness P'(0, 1, 2, ...m/? = P(0, 1, 2...m/g-P(0, 1, 2 ... m/(j- 1). Then for a quantic of the order m, the number of asyzygetic covariants of the degree and of the order /i is P'(0, 1, 2...m/i(m5-A*). In particular, the number of asyzygetic invariants of the degree 6 is P'(0, 1, 2...mf^e. To find the total number of the asyzygetic covariants of the degree 6, suppose first that md is even ; then, giving to /x the successive values 0, 2, 4, . . . mO, the required number is P{^e) -PQm^-l) + P(im5-l)~P(Jm^-2) + P(2) -P(l) + P(1) = P(W)> L e. when md is even, the number of the asyzygetic covariants of the degree 6 is P(0, 1, 2...m/imd; and similarly, when md is odd, the number of the asyzygetic covariants of the degree d is P(0, 1, 2, ...m)•i(»^5-l)• But the two formulae may be united into a single formula; for when inB is odd \md is a fraction, and therefore P{^ff) vanishes, and so when mB is even \{m6—\) is a fraction, and Pi(m5 — 1) vanishes; we have thus the theorem, that for a quantic of the order m\ The number of the asyzygetic covariants of the degree 6 is P(0, 1, 2...m)«im5 + P(0, 1, 2, ... m)«Kwid- !)• 41. The functions P{\m0\ &c. may, by the method explained in my "Researches on the Partition of Numbers," [140], be determined as the coefficients of a^ in certain functions of a? ; I have calculated the following particular cases : — Putting, for shortness, P'(0, 1, 2,... m)* ^m5 = coefficient of in <f>m, C. II. 34 266 A SECOND MEMOIR UPON QUANTICS. [141 then 4>2 = then ^ = 1 ^ (l-a;»)(l~aj»)' Afi - (1 -a? )(l+a?-a:'-a?*-- je'+jg^ + a:») '''^ "■ (l-««)«(l-a:»)(l-a;*)(l-a:») ' .Q^ (l--a?)(l+a?-d^~a;* + a^ + af + g» + g» + a^<>-g» + a;^» + a:^8) (l-a;»)«(l-a:»)«(l~j;*)(l-a:»)(l-a;') P(0, 1, 2, ... m)* ^md = coefficient of a:^ in -^m, then i/^2 = Vr3 = l+ar* (l~a;»)'(l-a;*)' ''' (l-a?)«(l-a;»)(l~a^)* 1 + a:* + 6a:* + 9^* + 12g» + Qx"^ -k- 6a^ + x"^ -^ a^"" P(0, 1, 2, ... m)*^md— 1) = coefficient of a:;* in -^m, ''^ (l-a»)«(l-a;*)' ''^ (l-a:»)»(l-«*)(l -«•)(! -ic') And from what has preceded, it appectrs that for a quantic of the order m, the number of asyzygetic covariants of the degree ^ is for m even, coefficient a^ in -^i, and for m odd, coefficient a^ in (yp^m + yfr^m); and that the number of asyzygetic invariants of the degree is coefficient a:^ in ifym. Attending first to the invariants: 42. For a quadric, the number of asyzygetic invariants of the degree is 1 coefficient of in l-a;»' which leads to the conclusion that there is a single irreducible invariant of the dijgree 2. 141] A SECOND MEMOIR UPON QUANTICS. 267 43. For a cubic, the number of asyzygetic invariants of the degree is coefficient a? in .j — -., 1 — ar i.e. there is a single irreducible invariant of the degree 4. 44. For a quartic, the number of asyzygetic invariants of the degree is coefficient of in (i _ J^^^ _ ^) . i.e. there are two irreducible invariants of the degrees 2 and 3 respectively. 45. For a quintic, the number of asjrzygetic invariants of the degree is l^af + a^ coefficient «• in (l-a;*)(l-ir«)(l-a^)' The numerator is the irreducible factor of 1 — ic", Le. it is equal to (1 — a;") (1 — «•) -r(l — a:") (1 — a:") ; and substituting this value, the number becomes 1 — ^ coefficient a:^ in (l-a;*)(l-a*)(l-a:")(l-d5»)' Le. there are in all four irreducible invariants, which are of the degrees 4, 8, 12 and 18 respectively; but these are connected by an equation of the degree 36, i.e. the square of the invariant of the degree 18 is a rational and integral function of the other three invariants; a result, the discovery of which is due to M. Hermite. 46. For a sextic, the number of asyzygetic invariants of the degree is /»• X a - (l'-x)(l'\-x — a^ — a^ — a:^ + w' + a^) coefficient ar in ^^ — /■. ^x, /■. ijwi zitti =;\ • (1 - a:;*)* (1 - a^) (1 - a?*) (1 - of) the second factor of the numerator is the irreducible fiswtor 1 — a:**, i. e. it is equal to (l-aJ»)(l-a:»)(l-a^)(l~a:»)4-(l-a:")(l-a^<>)(l-aj»)(l-a;); and substituting this value, the number becomes l-a:» coefficient as* in (1 - a:') (1 - a?*) (1 - a;*) (1 - x"') (1 - ai'') ' Le. there are in all five irreducible invariants, which are of the degrees 2, 4, 6, 10 and 15 respectively; but these are connected by an equation of the degree 30, i.e. the square of the invariant of the degree 15 is a rational and integral function of the other four invarianta 47. For a septimic, the number of asyzygetic invariants of the degree is l-a:«-f2a;»-a^* + 6a^ + 2a^* + 6a:^« + 2a:" + 5a:»-a;«+2a;^-a:« + a* coefficient a;* in (1 - a?*)(l ~a^)(l -a;«)(l -a;^»)(l - x"') 34—2 268 A SECOND MEMOIR UPON QUANTICS. [141 the numerator is equal to (1 - a:-)(l -a:»)-*(l -ir^»)(l -a:")-*(l -«")-^.... where the series of factors does not terminate; hence [incorrect, see p. 253] the number of irreducible invariants is infinite; substituting the preceding value, the number of asyzygetic invariants of the degree is coefficient a:^ in (1 - aj*)-> (1 - a^y' (1 " ^^ (1 - ^^ V- • • The first four indices give the number of irreducible invariants of the corresponding degrees, i.e. there are 1, 3, 6 and 4 irreducible invariants of the degrees 4, 8, 12 and 14 respectively, but there is no reason to believe that the same thing holds with respect to the indices of the subsequent terms. To verify this it is to be remarked, that there are 1, 4, 10 and 4 asyzygetic invariants of the degrees in question respect- ively; there is therefore one irreducible invariant of the degree 4; calling this Z4, there is only one composite invariant of the degree 8, viz. X/; there are therefore three irreducible invariants of this degree, say Xg, Xg', Xg". The composite invariants of the degree 12 are four in number, viz. X^*, X^X^, X^X^\ X^X^\ and these cannot he connected by any syzygy, for if they were so, X^\ Xg, Xg', X," would be connected by a syzygy, or there would be less than 3 irreducible invariants of the degree 8. Hence there are precisely 6 irreducible invariants of the degree 12. And since the irreducible invariants of the degrees 4, 8 and 12 do not give rise to any composite invariant of the degree 14, there are precisely 4 irreducible invariants of the degree 14 48. For an octavic, the number of the asyzygetic invariants of the degree is coefficient ar* in (1 -a;)(l -f x-a^- a^ + a^ + a?' -h a;* + ic* + a?** - a?*' - «*' -h «" + «*•) . {l-x'yil -a:»)«(l -a:*)(l -ar»)(l ^af) and the second factor of the numerator is (1 -a:)-»(l -a:»)(l -ar')-»(l -a:V(l -^)"' (l-af)-'(l -ar»V(l -a;**)(l -a?*0(l -^') - where the series of factors does not terminate, hence [incorrect] the number of irreducible invariants is infinite. Substituting the preceding value, the number of the asyzygetic invariants of the degree is roeffla;* in (1 - j;*)-' (1 - ar»)-i (1 - ar*)-^! - 4^)-> (1 - a:«)-i (1 - ar^-^ There is certainly one, and only one irreducible invariant for each of the degrees 2, 3, 4, 5 and 6 respectively; but the formula does not show the number of the irre- ducible invariants of the degrees 7, &c. ; in fact, representing the irreducible inva- riants of the degrees 2, 3, 4, 5 and 6 by X„ X,, X4, X5, Xg, these give rise to 3 com- posite invariants of the degree 7, viz. X,X^j, XjXj, X,X4, which may or may not be connected by a syzygy; if they are not connected by a syzygy, there will be a single irreducible invariant of the degree 7 ; but if they are connected by a sjzygy, there will be two irreducible invariants of the degree 7 ; it is useless at present to pursue the discussion further. \ 141] A SECOND MEMOIR UPON QUANTICS. 269 Considering next the covariants, — 49. For a quadric, the number of asyzygetic covariants of the degree is 1 coeflScient of in (\-x){\-<d^y i.e. there are two irreducible covariants of the degrees 1 and 2 respectively; these are of course the quadric itself and the invariant. 50. For a cubic, the number of the asyzygetic covariants of the degree 6 is coefficient of m ^^_Jy^^^^) • The first £Etctor of the numerator is the irreducible factor of l-a^, = (l-a^)-(l-fl?), snd the second factor of the numerator is the irreducible factor of l-aj*, = (l-aj*)^(l-a^); ^substituting these values, the number is coefficient of in (1 -a?)(l -a;»)(l -a:»)(l -iT*)' i.e. there are 4 irreducible covariants of the degrees 1, 2, 3, 4 respectively; but these ^re connected by an equation of the degree 6; the covariant of the degree 1 is the oubic itself U, the other covariants are the covariants already spoken of and repre- sented by the letters JJ, 4> and V respectively {H is of the degree 2 and the order 3, <X> of the degree 3 and the order 3, and V is of the degree 4 and the order 0, x.e. it is an invariant). 51. For a quartic, the number of the asyzygetic covariants of the degree 6 is coefficient sfi in (l-«)«(l-aj»)(l-aj»)* ^he numerator of which is the irreducible factor of l—af^ i.a it is equal to (1 — «•) (1 — «) -r (1 — oj*) (1 — aj*). Making this substitution, the number is 1 — ir* coefficient of in (l-a?)(l-a^)»(l-a^)«' X.e. there are five irreducible covariants, one of the degree 1, two of the degree 2, ^,nd two of the degree 3, but these are connected by an equation of the degree 6. *irhe irreducible covariant of the degree 1 is of course the quartic itself U, the other irreducible covariants are those already spoken of and represented by /, H, J, 4> ^"espectively (/ is of the degree 2 and the order 0, and J is of the degree 3 and ^he order 0, Le. / and J are invariants, H is of the degree 2 and the order 4, 4> xs of the degree 3 and the order 6). 270 A SECOND MEMOIR UPON QUANTICS. [141 52. For a quintic, the number of irreducible covariants of the degree is the numerator of which is (l+a?)»(l-a?H-2a;»H-ir«H-2a^H-S«» + ««H-5a;'H-a^H-8a;* + 2a^'H-a?" + 2a;"-a^ the first £Eu;tor is (1 — ai)~^ (1 — a^y, the second factor is (1 -^)(1 -a^)-«(l -aj»)-»(l -a:*)-^(l -af)'^(l -afiy{l-x'y(l''a*y(l^a^y(l--x'^)-^(l^a!''y^..., which does not terminate ; hence [incorrect] the number of irreducible oovariants is infinite. Substituting the preceding values, the expression for the number of the asyzygetic covariants of the degree 6 is coeff. ^in (l-x)-^ (1-a'«)-«(1 -a;»)-«(l -.d?«)-»(l -a?*)-«(l -«•)*(! - x^)^ (I - a*f (I -j^)i(l -ar»«)-«(l -^»)"**.... which agrees with a previous result: the numbers of irreducible covariants for the degrees 1, 2, 3, 4 are 1, 2, 3 and 3 respectively, and for the degree 5, the number of irreducible covariants is three, but there is one syzygy between the composite covariants of the degree in question ; the difference 3 — 1 » 2 is the index taken with its sign reversed of the factor (1 — aj")~*. 53. I consider a system of the asyzygetic covariants of any particular degree and order of a given quantic, the system may of course be replaced by a system the terms of which are any linear functions of those of the original system, and it is necessary to inquire what covariants ought to be selected as most proper to represent the system of asyzygetic covariants; the following considerations seem to me to furnish a convenient rule of selection. Let the literal parts of the terms which enter into the coeflScients of the highest power of x or leading coefficients be represented by Ma, Mp, My,,., these quantities being arranged in the natural or alphabetical order; the first in order of these quantities M, which enters into the leading coefficient of a particular covariant, may for shortness be called the leading term of such covariant, and a covariant the leading term of which is posterior in order to the leading terra of another covariant, may be said to have a lower leading term. It is clear, that by properly determining the multipliers of the linear functions we may form a covariant the leading term of which is lower than the leading term of any other covariant (the definition implies that there is but one such covariant); call this 0. We may in like manner form^ a covariant such that its leading term is lower than the leading term of every other covariant except 0i ; or rather we may form a system of such covariants, since if 4>, be a covariant having the property in question, <l)j + Ar0i will have the same property, but k may be determined so that the covariant shall not contain the leading term of 6i, i.e. we may form a covariant B, such that its leading term is lower than the leading term of every other covariant excepting 01, and that the leading term of ©i does not enter into 0,; and there is but one such covariant, 0,. Again, we may form a covariant 0, such that its leading term is lower than the leading term of every other covariant excepting 0i and 0,, and that the 141] A SECX)ND MEMOIR UPON QUANTIC8. 271 leading terms of Bi and B, do not either of them enter into B, ; and there is but one such co variant, B,. And so on, until we arrive at a covariant the leading term of which is higher than the leading terms of the other covariants, and which does not contain the leading terms of the other covariants. We have thus a series of covariants Si, B„ B,, &c. containing the proper number of terms, and which covariants may be taken to represent the asyzygetic covariants of the degree and order in question. In order to render the covariants B definite as well numerically as in regard to sign, we may suppose that the covariant is in its least terms (Le. we may reject numerical £Etctors common to all the terms), and we may make the leading term positive. The leading term with the proper numerical coeflScient (if different from unity) and with the proper power of x (or the order of the function) annexed, will, when the covariants of a quantic are tabulated, be sufficient to indicate, without any ambiguity whatever, the particular covariant referred to. I subjoin a table of the covariants of a quadric, a cubic and a quartic, and of the covariants of the degrees 1, 2, 3, 4 and 5 respectively of a quintic, and also two other invariants of a quintic. [Except for the quantic itself, the algebraical sum of the numerical coefficients in any column is =0, viz. the sum of the coefficients with the sign + is equal to that of the coefficients with the sign — , and I have as a numerical verification inserted at the foot of each column this sum with the sign ±]. ( Covariant Tables (Noa 1 to 26). No. 1. No. 2. 5 ^» yY' a+l 6 + 2 c + 1 =fc 1 The tables Noa 1 and 2 are the covariants of a binary qi^adric. No. 1 is the quadric itself; No. 2 is the quadrin variant, which is also the discriminant. No. 3. No. 4. H^. UY \^y i/Y ± 1 ± 1 ± 1 No. o. No. G. a-c/-f 1 a6rf+3 acrf-3 OiP -1 a6c~ 3 ac" -6 b*d +6 bed + 3 A» +2 b^c +3 bc^ -3 c» -2 5-, yr :L H :i^6 ± 6 ± 3 a'd' + 1 abed -6 a^ + 4 bH + 4 6»c« - 3 ± y The tables Noa 3, 4, 5 and 6 are the covariants of a binary cubic. No. 3 is the cubic itself; No. 4 is the quadricovariant, or Hessian; No. 5 is the cubicovariant ; No. 6 is the invariant, or discriminant. And if we write No. 3 = IT, No. 4 = if. No. 5 = *, No. 6 = V, 272 A SECOND MEMOIR UPON QUANTICS. then identically, No. 7. a+1 6 + 4 C4- 6 (^ + 4 « + l $«.y)* No. 8. 00 + 1 W- 4 c« + 3 No. 9. ( ±* oe + l 6«^1 ad + 2 6c-2 06+ 1 6rf+2 c«-3 6« +2 crf-2 6J+1 c« -1 $^y)* Jr 1 ±2 ±3 db2 ± 1 No. 10. ac6 + 1 OflP -1 b^e -1 bed + 2 c» - 1 No. 11. ±s o«rf+l o*« + 1 o56 + 5 oe« a(f0<- 5 a^ -1 6tf« - 1 o6c-3 o6i+2 aed^ 15 o^-lO 6c0 + 15 6<fe-.2 C6fe -f 3 6» +2 oc«-9 6y + 10 6»tf + 10 bd'-lO c"tf + 9 rf» -2 6»c +6 6c* bed er • . • <?d crf»-6 ^ ^8 :1:9 :k 15 dblO :l:16 :t9 ±8 No. 12. oV + 1 o«6c£e» — 12 oVe* — 18 a^cd^e -f 54 a^d^ — 27 ab'e^ + 54 ab'd'e — 6 abc^de — 180 abecP + 108 oc^ + 81 aA?d} — i 54 6V — 27 6'o<^ + 108 6»ci» — 64 6V« — 54 6V(i* + 36 6c*rf • • • c« • • • ^442 The tables Nos. 7, 8, 9, 10 and 11 are the irreducible covariants of a qu No. 7 is the quartic itself; No. 8 is the quadrinvariant ; No. 9 is the quadricova] or Hessian; No. 10 is the cubinvariant ; and No. 11 is the cubicovariant. The No. 12 is the discriminant. And if we write No. 7 = CT, No. 8 = /, No. 9 No. 10 = J, No. 11 = *, No. 12 = V, then the irreducible covariants are connected by the identical equation jir»-/f7«fr+4fr» + 4)« = o, and we have V = /»-27J«. 141] A SECOND MEMOIR UPON QUANTICS. 273 [The Tables Nos. 13 to 24 which follow, and also Nos. 25 and 26 which are given in 143 relate to the binary quintic. I have inserted in the headings the capital letters A, B, . . . L and also Q and Q' by which I refer to these covaiiants of the quintic. A is the quintic itself, C is the Hessian, G is the quartin variant, J a linear covariant: Q is the simplest octinvariant, and Q' is the discriminant. As noticed in the original memoir we have AI + BF-CE = 0; and Q' = G« - 128 Q, only the coefficient 128 was by mistake given as 1152.] A. No. 13. ( a + 1 6 + 5 c+10 rf+10 6 + 5 /+i Xa^y)' B. No. 14. ( ae+1 bd-i c* +3 a/+l 6e-3 ed+ 2 kf+1 ce-4: d»+3 tn^ y)* :1:4 :1:8 ±4 C. No. 15. ( oc + 1 6* -1 ad+3 be -3 ae + 3 W+3 c" -6 (l/'+l 6« +7 crf-8 6/ + 3 cd +3 cP -6 C/ + 3 de-Z d/^l e» -1 H^ yY db 1 =k8 :t 6 ^S ±6 D. No. 16. ^B =fc 1 ( a^ ... 06/"+ 1 a4f*\ o«^ ... ace + 1 ode— 1 w? -1 6<ir+i atP-1 jy-i 6c^-l W -1 Va -1 &C0 + 1 5<2«-i-l «y-i hcd+2 M»+l «»« +1 ed0 + 2 c* -1 c'rf-l aP-l <i» -1 $«, y)' :^8 8 8 8 K No. 17. ay+1 ay+ 5 acf-^ 2 adf- 2 cw/- 5 qr- 1 a6d - 5 ace - 16 ocie-12 oa* - 8 64/'+ 16 6i2/-+5 , 0(^+2 V 6*^ + 8 a<i'+ 6 6!/'+ 8 6c/ +12 6«« + 9 c(i/'-2 6»6 - 9 6c« -38 6(20+38 c*/- 6 C6* -8 he -6 5c(2+38 M> + 72 i?e -72 ce20-38 (^6 +6 c» -24 c'rf-32 c«P + 32 d» +24 :1:11 :t49 :1:S2 :1>S2 :i:49 dbii $^. yy F. No. 18. I (i»rf+l a6c-3 6» +2 aV + 2 a6<;+ 1 oa" -12 6»c + 9 a^f^ 1 a6a + ll occf- 34 6»rf+16 6c» + 6 a6/+ 7 ace — 8 arf«-34 6*6 +29 bed- 2 c* + 8 a^+ 5 acfo-40 6*/ + 16 6c6 + 47 6c^-44 ed + 16 o^/-- 5 ac« -16 6c/ + 40 bde-^7 &e +44 c(P -16 oc/- 7 6(f/+ 8 6«» -29 c«/ + 34 e£f0+ 2 d» - 8 <!/••- 1 6e/-ll «i^+34 (»» -16 cPtf - 6 6/«- 2 <»/- 1 rf»/+12 (fo« - 9 C/^-1 cis/'+S e» -2 :l:8 :& 12 C. II. :k84 :1:44 :k84 84 ^44 :S=84 ^\2 35 $«^y)' 274 A SECOND MEMOm UPON QUANTICS. [141 G. No. 19. «!/•*+ 1 a6(?/- 10 acdf-^ 4 occ* + 16 ad^e- 12 hHf^ 16 6V + 9 6cy -12 hcde - 76 hd" +48 c^tf + 48 c^d^ - 32 H. No. 20. ^ 142 a«4/'+ 1 aV + 2 ay « + 1 a6/«+ 3 a^«+ 1 a V a6^- 4 a6«/ - 4 acef- 4 adef- 3 aftc/- 3 aM -10 occj/'- 2 aciy- 2 a«« + 2 aftc2s - 5 ac»/- 2 ace" + 4 oc^ + 4 6y« ... ao*« + 10 oecfe + 24 a^i'e 6V - 10 beef - 5 / acd^ — 4 0(^-12 ft«rf/+ 4 6ai^+ 24 bd"/-^ 10 ^ 6'/ + 2 6»</ + 4 6V - 9 ftc«» + 16 M«» - 5 h^ce - 5 W« +16 W/ ... 6c^e -22 c»4/-- 4 6«<^ +14 ftc«e -22 6ccfo + 50 cy -12 c*fi" + 14 he'd - 16 6cc^ - 4 W -36 c«<fo - 4 crf«« - 16 c* + 6 i c«rf + 8 c»« -36 cW +28 cc^ + 8 rf* + 6 dbSS db54 j=87 db54 =k83 5*. i^)* I. No. 21. ttV* + 1 a«ei/+ 2 a*^ ay« ... abp .., ac/«- 2 «(/••- 1 a*(ie - 1 aV - 2 abd/ + 2 cU>e/ ac«/-2 culef a^y +1 ' a6y-l a^/-10 a6e" -2 acdf ... acV+l ae> + 2 her + 1 abce - 2 a6c^+ 10 ocy-l a<j«» -20 arf(5» + 1 6y» + 2 6cfe/+2 a6«/* + 4 ac** acde-2 arf*« + 20 6y +2 beef bSf-^ 10 6«* -3 . ac*rf - 1 acd^ cuP +S b^d/ + 20 M^+2 c V - 4 < 6»e +3 6y - 2 b'cf - 1 6V 6<w* -5 6cfe« -14 «P/ + 1 b^cd -6 6»c« + 14 b^de +6 W/ -20 bd'e - 1 c«(y-10 ocfe* +6 ftc* +3 b^d' + 2 6c»«5 +1 &C(i0 cy -3 c*«» - 2 (i»« -3 W(/ -26 6a^ -9 6rf» -20 c»ci(5 +9 ccTe + 26 1 c* + 12 c*rf +4 c»e +20 cd" -4 dl" -12 i 11 =k40 dbi5 =b0O J. 15 db40 dbll 5«, y)'- J. No. 22. o»c/« + 1 a»4/"»+ 1 a'def- 2 aV/ - 1 aV + 1 o6c/»- 2 ab*/*- 1 abdef- 4 aJc?/"- 4 o&e' + 6 aM'/+ 8 ac"*/" + 8 tM^- a aaPf- 2 ocM^- 2 a«fe'-12 w»f + 14 ad*e + 6 . acd»« - 22 < ad* + 9 *»/• + 1 6»«/ - 2 6V + 6 6V/ + 14 Vecif-\2 VdU? -16 6'c«« -16 fte»4r-22 iWj +10 6«V +10 ftc*/ + 6 iedV +30 6e'<2« +30 W -16 6c«i» -20 c*/ + 9 «rt» -16 «:•«*» -20 c»d' +10 «•«<» +10 \^yY :1:95 db95 141] A SECOND MBMOIB UPON QUANTICS. 275 K No. 23. aV* ... a\p + 1 a*e(/^- 1 aV' •• c^cef + 1 a*e^/- 5 aV/ + 1 aMf-- 1 a*cy- 3 aV + 4 €^P + 5 aW/+ 1 a»^ + 2 a^/«- 1 cMef-- 8 acy«+ 3 o^V- 1 ahcef + 8 a6e» + 3 (Kidef" 14 €^cdfAr 14 a6i«/+ 11 ocV-H ace* + 8 a^e» - 11 aW6«- 17 ocrfy+ll ad^f + 9 oftrf"* - 1 ac«(^- 11 (kcd^ + 6 ewTe* - 6 ocy - 9 ac»e» -16 oc^e - 6 W^ " 2 . ac*flfe 4- 14 V occP - 6 cuid^e + 44 6»/» - 4 h'def + 11 oc^* - 18 ft*ce/ 4- 17 6V - 9 h^df - 8 ft»6/ - 3 5«rf"/ + 16 hc'ef + 1 ft»(5» + 9 h^cdf ^ 6 h^d^ - 21 fccrfy - 14 ft*cy + 6 6«c«> +21 6c»e(/'-44 hcde" + 16 6^cei« - 16 6We - 6 W(5» + 5 5rf»« - 3 6»rf» + 8 6cy + 6 hcd^t + 39 c*^/' + 6 6c»d + 3 h<?dt - 39 5^/* - 12 C»6» - 8 hc'iP - 2 6crf» + 22 cy + 18 c»cP« + 2 <?*(/ cV + 12 c»rftf - 22 cc?* c»rf» - 8 c«(^ + 8 5^, yY ±57 ±139 ±129 ±67 L. No. 24. ( a^hdf ... aV/ + 2 a'cc2e — 5 aV +3 ai^cf --^ at^de + 6 aJlx^e + 5 abed:'''! a^d + 1 6*/ +2 6»c<!J -5 ft»(f« -2 6 Vrf + 8 6c* -3 «•/• ... a*6^ ... c^cdy-k- 7 a^ce" - 10 aWtf+ 3 a6^- 7 a^«» + 10 oftcy- 7 O^CG^tf— 8 rtW» + 9 oc^tf +22 ac«rf»- 19 ft*c/ + 7 6»dd + 2 W« - 19 bi^d +33 c» -12 aV' ... a\?«/"- 3 (^dy+12 a«<fo«- 9 a6V+ 3 a6c^-18 a6c6"- 18 aW*e + 30 w?f - Z 00*6^ + 45 a«i» -39 hi'df -• 6 6V +27 6*cy + 16 6»afo - 87 6^(^ + 6 Wtf +12 6c>(i« + 57 c« -24 a*c/»- 1 a^def-h 7 aV - 6 aiy»+ 1 abce/-26 aWy+32 a6<fo«- 8 ac«4/*- 18 ac«e* + 6 euxPe-^62 ad^ -39 l^ff +19 6«c<(/' - 53 6«ce» +20 ft'ci'tf -25 h^f +39 W<fe - 45 hcd^ +65 c*e c»(P -20 a«c(r+ 1 aV/- 1 a6^- 7 abde/-\'26 cM -19 oc'e/ - 32 acdy+18 aec;^ + 53 a^e -39 6»/« + 6 6»c«/+ 8 ft*rfy- 6 6«<^-20 Mdf^^h We» +25 6crf'tf-52 6rf* c*/ +39 c»(fo -65 c>rf» +20 aV' ... a6<(/^+ 3 oW/- 3 acy*-12 ac56/'+18 occ* + 6 a^f + 3 ad'e'-U 6»^«+ 9 6»^+18 6V -27 bc'ef - 30 6crfy-45 6ccfo*+87 c*cP« erf* -12 + 39 - 6 -57 + 24 ay» ... a6«/^ ... ac4/^- 7 ace*/ -h 7 a(Pe/+ 7 ode" - 7 6*rf/» + 10 ft'ey-lO bc'r- 3 6cd;5/'+ 8 6c<5» - 2 bd"/ - 22 6rf»e* + 19 C»6/ - 9 c»rf'/+19 c'de' +11 cciP« -33 <f +12 abp ... ac«/*" ... arfy-2 ade^/ + 4 cw* - 2 6'(5/* ... 6c«J/^ + 5 6cey -5 bd'ef -5 We» +5 cy> -3 C«rf6/ + 7 c«e» +2 crfy -1 cd»e» -8 ci*« +3 5«» y)' 26 93 ±207 241 ±241 ±207 ±93 ±26 No. 26, Q = + 1 o»cd/' + &C. No. 26. 35—2 276 [142 142. NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [Now first published (1889).] In the present paper I arrange in a more compendious form and continue to a much greater extent the tables (first of each pair) given Nos. 35 — 39 of my Second Memoir on Quantics, 141, pp. 260 — 264, which relate to the cubic, the quartic and the quintic functions; and I give the like tables for the sextic, the septimic and the octavic functions respectively. The cubic table exhibits the coefficients of the several xz terms of the function 1-t-(1— ^.1— a?-2r.l— a^^.l— a^z\ or, what is the same thing, it gives the number of partitions of a given number into a given number of parts, the parts being 0, 1, % 3, (repetitions admissible) : or again, regarding the letters a, 6, c, d, as having the weights 0, 1, 2, 3 respectively, it shows the number of literal terms of a given degree and given weight. And similarly for the quartic, quintic, sextic, septimic and octavic tables respectively, the parts of course being 0, 1,... up to 4, 5, 6, 7 or 8, and the letters being a, &, ... up to e, /, g^ h or t. The extent of the tables is as follows: cubic table extends to deg*weight 18 — 27 quartic quintic sextic septimic octavic it 18—36 18—45 15—45 12—42 10—40 viz. for the quintic, the sextic and the octavic functions these are the deg-weighta of the highest invariants respectively. I designate the Tables as the od-, a^-, a/"-, og-i ah' and ai-tables respectively. It is to be noticed that in the several tables the lower part of each column is for shortness omitted ; the column has to be completed by taking into it the series 142] NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICa 277 of bottom terms of each of the preceding columns: thus in the af- or quintic table the complete column for degree 3 would be D 8 W 8-7 ■V O 6 — 1 6 - 8 5 — 8 4 — 4 3 ■V n 2 — 6 1 — 7 1 where the concluding terms 2, 1, 1 are the bottom terms of the three preceding columns respectively. And the meaning is that for degree 3, and weight 8, or 7, the number of terms is = 6 ; for weight 7 — 1, =6, the number of terms is =» 6 ; and similarly for weights 5, 4, 3, 2, 1, the numbers are 5, 4, 3, 2, 1, 1 ; the numbers are those of the terms W. 8 8 at c?h a'e a*d aU «y abf acf adf <!&' abe abd ahe ace ode tuf 6' (U? acd ad? hV be/ 6'c l?d 6V b*ce bde he bed bcP <?d No. 8 The like remarks and explanations apply to the other table& D 1 8 8 4 6 6 7 8 od-TABLE. 9 10 u 18 18 14 16 16 17 18 W 8-18 6-4 6 8-7 9 11-10 18 14-18 16 17-16 18 80-19 81 88-88 84 86-86 87 1 1 2 LI. 3 3 5 4 4 6 6 5 8 8 7 7 10 10 9 8 1. 13 12 12 11 10 15 18 15 18 14 17 13 17 12 15 21 21 20 19 18 16 25 24 24 23 22 20 19 28 28 27 26 25 23 21 32 32 31 31 29 28 26 24 36 36 35 34 33 31 29 27 41 40 40 39 38 36 35 32 30 45 45 44 43 42 40 38 36 33 50 50 49 49 47 46 44 42 i 39 i 37 1 — 1 - 8 1 >4 14 — 6 * — 6 — 7 » 278 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [142 D 12 8 4 6 6 8 (w-TABLE. 10 11 la 18 14 16 16 17 18 W 8 4 6 8 10 12 14 16 18 20 24 26 84 86 113 5 8 12 18 24 33 43 55 69 86 104 126 150 177 207 241 1 1 2 4 7 11 16 23 31 41 53 67 83 102 123 147 174 204 237 ~1 2 4 7 11 16 23 31 41 53 67 83 102 123 147 174 204 237 3 5 9 14 20 28 38 49 63 79 97 118 142 168 198 231 5 8 13 19 27 36 48 61 77 95 116 139 166 195 228 6 10 16 23 32 43 56 71 89 109 132 158 187 219 9 14 21 30 40 53 68 85 105 128 153 182 214 11 17 25 35 47 61 78 97 119 144 172 203 [ 15 22 32 43 57 73 92 113 138 165 196 18 26 37 50 65 83 104 127 154 184 23 33 45 60 77 97 120 146 175 27 38 52 68 87 109 134 162 ■ 34 46 39 62 53 47 80 70 6S 101 90 82 125 113 104 153 139 129 1 54 71 64 92 83 72 116 106 93 -0 -1 -2 -8 -4 -6 -6 -7 -8 -9 -10 -U -12 -U -14 -16 -16 -17 -U a/-TABLE. D W 1 8 8 10 11 12 13 14 16 16 17 18 8-2 6 8-7 10 18-12 16 16-17 20 28- 26 28-27 80 88-82 86 88-87 40 43-42 46 1 1 3 6 12 20 32 1 3 6 11 19 32 2 5 11 18 30 2 4 9 16 29 3 8 14 25 6 11 23 5 9 1 7 19 16 12 10 49 48 46 43 39 35 30 26 21 17 13 73 71 70 66 63 57 52 45 40 33 28 22 18 102 141 101 141 98 137 93 134 88 127 81 121 74 111 66 103 58 50 43 35 29 23 92 83 72 63 53 45 36 30 190 188 184 178 170 161 150 139 126 114 101 89 77 66 55 46 37 252 249 247 240 233 222 212 197 184 168 154 137 123 107 94 80 68 56 47 325 322 317 309 299 286 272 256 239 220 202 183 165 146 129 112 97 82 69 57 414 414 408 402 390 379 362 346 325 306 283 262 238 217 194 174 152 134 115 99 83 70 521 518 511 501 488 472 453 433 409 385 359 333 306 280 253 228 203 180 157 137 117 100 84 649 645 641 630 619 601 583 559 536 507 480 448 419 386 356 324 295 264 237 209 185 160 139 118 101 795 791 783 770 754 734 711 684 655 623 590 554 519 482 446 409 374 339 306 273 243 214 188 162 140 119 967 966 957 948 930 912 886 860 827 795 756 719 677 637 593 553 509 469 427 389 350 315 279 248 217 190 163 141 142] NTJMBBICAL TABLES SUPPLEMENTABY TO SECOND MEMOIR ON QU ANTICS. 279 agf-TABLK D W 18 8 8 10 11 la 18 14 16 8 6 9 la 16 18 SI 84 8T 80 88 88 89 43 46 1 1 4 8 18 32 58 94 151 227 338 480 676 920 1242 1636 1 3 8 16 32 55 94 147 227 332 480 668 920 1232 1635 1 3 7 16 30 55 90 146 221 330 471 664 907 1226 1617 3 7 14 29 51 88 139 217 319 464 648 896 1203 1601 2 5 13 25 48 81 134 205 310 446 634 870 1182 1565 4 10 23 42 76 123 196 293 431 608 847 1145 1533 3 9 19 39 68 116 182 280 408 587 813 1113 1483 6 16 32 61 103 169 258 387 553 780 1064 1435 5 12 28 52 94 152 241 359 525 737 1021 1373 10 22 46 81 139 218 335 488 699 965 1316 7 18 37 71 121 199 304 455 650 914 1244 13 31 59 107 175 278 415 607 852 1178 11 24 51 91 157 248 382 557 798 1102 19 40 78 134 222 341 512 733 1031 14 33 25 64 54 117 98 193 170 308 271 462 419 677 614 952 882 20 42 [ 34 83 67 144 124 240 206 371 331 559 499 803 734 26 56 43 103 86 180 151 289 253 449 394 661 596 ( 35 69 57 129 106 216 187 349 302 529 472 44 88 70 156 132 263 223 412 362 58 108 89 192 159 303 270 71 134 109 228 195 90 161 135 110 1 3 8 4 6 6 7 8 9 10 11 18 18 14 16 16 17 18 19 80 81 84 86 -87 39 80 280 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [l42 oA-TABLE. D W 1 8 8 10 u la 4^ 7 11-10 14 18-17 n 86-M 28 88-81 86 88-88 48 1 1 4 10 24 49 94 169 289 468 734 1117 1656 1 4 10 23 48 94 166 285 464 734 1109 1646 1 3 9 23 46 90 162 282 454 722 1093 1634 ' 3 8 20 43 88 155 272 441 709 1069 1605 2 7 19 39 81 146 263 424 686 1038 1572 2 5 16 35 76 136 247 403 663 1000 1524 4 14 30 68 125 233 379 629 957 1475 3 11 26 61 112 214 354 598 908 1410 9 21 52 100 197 325 558 856 1346 6 17 46 87 176 297 520 799 1271 5 13 37 75 158 268 477 742 1197 10 31 63 137 239 437 682 1114 7 24 53 120 210 392 623 1036 19 42 101 184 353 563 950 14 34 86 157 311 506 871 11 26 20 70 58 134 112 274 236 449 397 788 711 15 45 36 93 75 204 171 346 300 633 564 27 61 145 256 493 21 47 37 119 98 218 182 432 372 28 78 63 152 124 320 270 48 101 229 38 80 64 189 157 « 49 127 103 81 65 11 18 18 14 16 18 17 18 19 -81 -84 -86 -88 -87 89 80 NUMERICAL TABLES SUPPLBBiENTABY TO SECOND MEMOIB ON QUANTICS. 281 W ai-TABLE. 1 8 8 4 6 6 7 8 9 10 4 8 18 16 80 84 88 38 86 40 ! . 1 5 1 4 13 12 33 31 73 71 151 147 289 285 526 519 910 902 1514 1502 — — 1 1 4 12 31 70 146 282 515 894 1492 — 8 1 3 11 28 66 139 272 499 873 1460 — 8 3 10 27 63 134 263 486 851 1430 — 4 2 8 23 57 123 247 461 816 1379 — 6 2 7 21 52 116 233 440 783 1331 — 6 5 17 45 103 214 409 738 1265 — 7 4 15 40 94 197 383 696 1214 — 8 3 11 33 81 176 348 645 1127 — 9 9 28 71 158 319 597 1057 -10 6 22 59 137 284 543 974 -U 5 18 51 120 255 495 900 -18 13 40 101 221 441 816 -18 10 33 86 194 394 742 -14 7 25 20 70 58 164 141 345 302 662 593 -15 -16 14 45 116 258 519 -17 11 36 27 97 77 222 185 457 393 -18 -19 21 63 156 340 -80 15 48 38 127 104 286 243 -81 -88 28 82 200 -88 22 66 50 167 134 -84 -80 39 109 -86 29 85 68 -87 -88 51 -89 40 -30 ^he numbers of each table are connected in several ways with those of the ling tables. One of these connexions, which is of some importance, is best ex- d by an example: in the a/*-table, 8-80, the number of terms of degree 8 and t 80 is 73 ; and we have 73 = 1 -h 6 + 16 + 23 + 27. viz. (see p. 288) these le numbers of the terms in a*, a', a", a\ a* respectively: the complementary 3, (for example) of a' are be/*, &c. terms in 6, c, d, e, f of the degree 5 and t 80, and (replacing therein each letter by that which immediately precedes it) are in number equal to the terms in a, 6, c, d, e of the degree o and weight = 15 ; thus the number 6 of the terms in question is that for the deg- weight f the a6-table: and so 1, 6, 16, 23, 27 are the numbers in the o^-table for eg-weights 4-i6, 5-i8, 6-14, 7-i8 and 8-i8 respectively, or (making a change rendered lary by the abbreviated form of the tables) say for the deg- weights 4-o, 5-io, 18 and 8-18. •1 II. 36 282 [143 143. TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUIN TIC: FROM THE SECOND, THIRD, FIFTH, EIGHTH, NINTH AND TENTH MEMOIRS ON QUANTICS. [Arranged in the present form, 1889.] The binary quintic has in all (including the quintic itself and the invariants) 23 covariants, which I have represented by the capital letters, A, B, C, . . . W (alternative forms of two of these are denoted by Q' and S'). The covariants A, . . . L, and also Q, Q' were given in my Second Memoir on Quantics, and except Q and Q' arc reproduced in the present reprint thereof, 141 ; in all these I gave not only the literal terms actually presenting themselves, but also the terms with zero coeflBcients; in the other covariants however, or in most of them, the terms with zero coefficient*^ were omitted. It is very desirable to have in every case the complete series of literal terms, and in the covariants as here printed they are accordingly inserted: the number of terms is in each case known beforehand by the foregoing q/*-table, 142, and any omission is thus precluded; by means of this ci/*- table we have the numbers of terms as shown in the following list. I have throughout (as was done in the Ninth and Tenth Memoirs) expressed the literal terms in a slightly diflferent form from that employed in the Second Memoir: this is done in order to show at a glance in each column the set of terms which contain a given power of a, and in each such set the terms which contain a given power of b. The numerical verifications are also given not only for the entire column but for each set of terms containing the same power of a; viz. in most cases, but not always, the positive and negative coeflScients of a set have equal sums, which are shown by 43] TABLES OF THE COVABIANTS M TO W OF TH^: BINARY QUINTIC. 283 number with the sign ± prefixed. The verification is in some cases given in regard I the subsets involving the same powers of a and b, here also the sums of the )sitive and negative coefficients are not in every case equal. The cases of inequality ill be referred to at the end of this paper. The whole series of covariants is as follows : Hem. 2 No. of table 13 • A ^ (1. 1. 1, 1, 1, IJx, yy (leg-weight 1 (0....5) »» 14 B ^ (3. 3, 35ar, y)» 2 (4.6) » 15 = (2, 2. 3. 3. 3, 2, ijx, yf 2 (2 8) 11 16 D ss (6. 6, 6, 6$». yy 3 (6.. 9) *i 17 E = (5. 6. 6. 6, 6, 6$«, yy 3 (5. ...10) it 18 F = (3, 4. 5. 6. 6, 6, 6, 5. 4, 3$a:. y)» 3 (3 12) 1* 19 G = (12$a;, yy, Invt. 4-10 M 20 U ^ (11, 11, 12, 11, 11$^, yy 4 (8., .12) )) 21 I ss (9, 11. 11, 12, 11, 11, 9$a!, yy 4 (7 13) »» 22 J s (20, 20$*, yy 6 (12, 13) »» 23 E = (19, 20, 20, 19$«. yy 5 (11.. 14) »» 24 L = (16, 18, 19, 20, 20, 19, 18, 16$ir, y)' 5 (9 16) 8 83 M = (32, 32, 32$«, yy 6 (14 . 16) >i 84 N = (30. 32. 32, 32, 30$a;, y)« 6 (13.. .17) 9 90 = (49, 495a!, yy 7 (17. 18) »> 91 P = (46, 48, 49. 49, 48, 46$*, yy 7 (15.. ..20) 2 Q 25 Q'26 Q.Q' t (73$«, yy, Invt. 8-20 9 92 R =5 (71. 73, njx, yy 8 (19.21) 9 10 S93 S93&W S,S' = (101, 102, 102, 101$ar, y)» 9 (21.. 24) 9 94 T S (190, 190$ar, y)' 11 (27, 28) 3 29 U = (252$ar, y)», Invt. 12-30 9 95 V = (325, 326$ar, yV 13 (32, 33) 5 29a W = (967$a!, y)», Invt. 18-45 36—2 284 TABLES OF THE OOVARIANTS M TO W OF THE BINARY QUIMTIC. [Ud M. No. 83. cflfief • ■ • a» 6y • • • a* b^f* a' 6W» • • • o» 6y • • • b'eef* «•/ • • • Ved^* - 1 dy - 1 fiV/* - 1 c^f + 1 A»/ + 2 edef + 6 dV" + 1 «* - 1 of - 3 d^ - 1 a' ft««/» dff - 3 a» m/* + 1 b'ed/* + 6 «?"«• + 2 «•/ - 1 <!«■/- 5 a' 6»c/» + 2 6'c«/« + 1 ««V - 6 A/ - 5 cdef + 6 <fc» + 6 «» + 3 of - 8 bvy - 3 6V?/" - 5 dPf -10 c»«fc/ + 7 ccPf + 7 d^if + 11 «•«» + 2 ed<? - 1 bVef -10 aP/ - 1 dP« - 1 <?dff + 11 «?«■ - 8 h*(?dff - 1 <?d^ + 18 <?« + 3 <?^ + 6 ecPe -28 a^6«i^ - 3 <?dPe - 8 (P + 9 «!/• + 3 evf + 3 a<> 6»c/» - 1 6V/» + 2 a* b*/* - 1 def - 8 afo/ - 1 Vcrf + 5 «> + 9 «»» - 3 <Pf + 2 l^«/ + 11 dy + 6 d^ - 3 etPf + 18 «?«■ - 4 Wdf - 8 cdt? -37 5'c««/ - 1 «»«■ - 4 <?« + 8 c'd'/ - 8 cePs + 7 b'<*d/ -28 cVfc* + 7 # - 1 c»«» + 8 wPe + 5 6"cy + 3 c'd'e + 37 <i» - 3 cida + 6 cd^ -17 b'e'd/ + 3 i?df - 4 6V/ + 9 C«(J» - 1 6V« - 3 c^ -17 c»<?« - 4 c^d* - 2 e'd' + 8 <!«# + 2 5*, y)^ :fc 7 SI 84 2 67 106 ^ 2 22 28 ^ 62 db 167 db 62 143] TABLES OF THE 00VARIANT8 M TO W OF THE BINARY QUINTIC. 285 N. No. 84. a»6«(^ - 1 a» 6»«/» a« 6y» a« 6y» a« 6 V* + 1 ^/ + 1 a> h^df^ ~ 4 a« 6»(5/» 6»c«/« + 4 deP - ^ 1 a» h'cf^ + 3 «y + 4 6»c(^ rfy* - 4 «y + 2 def + 2 6V/» + 4 c«y + 6 (i^y - 4 a* 6y» - 1 <5» - 5 edef - 8 d««/ - 12 «< + 4 h'ceP - 2 6V^ - 8 c«» + 4 cie» + 6 ai 6V' - 4 d'P + 8 cdy + 2 d^f a' 6"(^« - 6 6*c(^« + 8 cfey - 2 cd^ + 12 cP«* ey c«y - 16 «* - 6 cP« - 6 a* 6"c/» + 4 6»(y« + 12 d'ef + 48 6 V(^ - 2 a^fty* - 2 def + 16 cdef dfe» - 32 c»«y + 6 fe'c^/ - 2 «> - 24 ce» - 36 6V/« cd^ef- 20 ^/ - 6 h't^ef - 48 cPf + 48 c«cfe/- 40 cdfe» + 12 d^ + 13 ccP/ + 40 d«c« - 12 cV + 56 rfy + 9 6ic«e(/" + 20 ccfe» + 40 6Ve/ - 48 cd^f + 8 rf»«« - 6 c«c« + 4 d»« - 24 c*gP/ ... c(?e» - 40 a» 6»(5/* + 5 CflPc - 52 h^(*df - 8 <?d^ + 156 rf*e + 12 6«c(^« - 12 rf* + 24 c»c« + 56 ccPc - 168 a» 6»(^« - 4 cey - 13 6V/ - 9 c>(?tf - 88 rf» + 54 ey + 24 (/»«/ - 4 C»(fe + 20 c^ + 36 a« 6»c/» - 6 6"cy» d8» + 15 c*rf» - 10 ao 6y« - 4 def + 36 cflfe/" - 40 6V/« + 6 «*• 6 V + 6 h'cef + 32 o • • . ce» - 60 c«cfe/+ 52 l^cdf + 12 d«/ - 56 6V«/ + 12 dy - 56 c«c» - 10 C6« - 15 cfe> + 60 cd^f - 156 d»e» + 100 cdy - 20 rf*c + 10 b'i^df + 40 ccfe* 6Vc/ + 24 c(i»«« - 30 6V/ + 6 c>e» -100 d^e + 60 c«dy+ 88 d^e + 15 c*cfc + 30 c<?e - 80 6»c»e(/" + 168 c«rfe» + 80 6Ve/ - 24 cd» - 20 rf* + 60 c'c' - 60 ciPe - 200 c»rfy+ 10 i^c^c - 15 6V/ - 12 c'cPe d" + 60 c»cfe> + 20 c»rf» + 10 (*de + 200 c(^ - 30 6Ve(r - 36 <?d^e - 10 6Vrf • • • c«rf» -120 6V/ - 54 c*6» - 60 ccP 6Ve - 60 c*(ig + 30 c^d^e + 120 c*(? + 40 c'cP cW - 40 di 1 19 db 12 :i= 12 db 8 db 8 81 192 270 182 87 62 482 806 496 128 \^^ yY ± 168 db 686 d. 588 d= 686 db 168 286 TABLES OF THE 00 VARIANTS M TO W OF TUB BINABT QUINTIC. [143 0. No. 90. a» 6V/» + 1 a» b^dp - 1 dtp - 4 «y* + 1 «y + 3 a« 6V' + * a« 6"/» - I (fo/* + 3 b'ceP - 3 ^/ ~ 7 d'P + 16 b^t^eP - 16 cfey + 4 ccr/*+ 6 «* - 15 c<foy+ 30 6 V(^« - 6 ce* - 8 c»cy + 4 JV - 18 c<i»c/- 22 d»«' + 6 cdg> + 26 a* 6»/» - 3 rf*/ + 9 il'ceP - 4 d»c« - 12 (?/« - 4 a» b^eP + 7 cfey - I 6«<^ - 30 tf* + 18 c^f + 1 6^(4/^+ 22 rf»c/ - 74 c»e»/+ 74 cfc» + 84 c<^«/-160 6'cy + 18 «fo» - 32 c'def + 160 rf*/ + 81 c'tf' - 98 d»e» + 6 e(P/ - 20 fcV/* - 9 cd'e' - 94 (^de/^ 20 c?'^ +51 c»«» -112 6Vc/ - 81 c«dy- 18 c»dy+ 18 c«rf»«» + 284 i^de" + 140 cd'e -216 C»d»<5 - 100 d» + 54 crf» + 18 a*6*c/« + 15 a» b'dP + 8 b^cdp - 26 e»/ - 18 ce'/ - 84 b'c'P - 6 (?»«/ + 98 crife/" + 32 d^ 45 c<5» + 45 6V/« + 12 d»/ + 112 c'de/-^ 94 (i»e» - 150 c««' + 150 6V«/ - 6 cd*/ - 140 c>rfy - 284 ccP«« - 50 c»cfe> + 50 d'e + 15 ccPe + 320 b'c'ef - 51 rf» -120 c>d«/ + 100 6»c*ej^ +216 c»cfe« - 320 cV - 15 cW« + 310 c«(i»« - 310 crf» - 90 e'd' + 130 6Vrf/ - 18 ftocy - 54 c»«« + 120 c^de + 90 c*(?« - 130 c*d» - 40 c»rf* + 40 :&: 4 ± 1 69 49 497 669 1003 954 H^. yY 1563 dbl563 8] TABLBS OF THE COVARIANTS M TO W OF THE BINABT QUINTIC. 287 P. No. 91. • • • a» Vf* a»6V' - 1 a« h^df^ + 1 a» b^eP • • • c^ b^P • • • VceP - 2 deP + 6 ^P - 1 a* bHp + 2 a^b^ep + 1 dy« + 6 ^/ - 5 a« b'cP - 6 cy« - 2 b^cdp - 1 - 2 d^f - 1 a« 6y« + 1 d^ + n b^^p - 5 c«y+ 1 + 2 tf* - 2 6^C<5/« - 11 ey - 5 edeP^ 17 (PeP+ 3 - 1 a* \?eP + 2 d^P - 4 6Ve/« + 4 c^f - 7 flfey - 5 - 1 h^cdp - 17 (^/ - 4 cd'P- 2 d^P -^ 4 e» + 2 + 2 c^f + 13 c* + 17 c(iey+ 4 d^e^f- 6 a} b^df^ + 2 - 3 ^tf - 32 y^&dp^ 2 cc* — 4 d^ + 5 ey« - 2 - 6 cfe» + 32 c»«y + 26 d'ef - 10 a* 6V* + 1 h'ifp - 2 + 13 6V/« + 4 c^ef- 2 rf'tf' + 8 cfo/« - 13 cdtp^ 6 - 8 c»cfe/+ 36 cd^ - 40 a' 6y» + 5 ^/ + 12 cey - 2 + 2 c»«» - 24 rfy - 9 b^cep + 4 6'cV^ + 32 d^P - 16 + 16 cd^f - 10 rPe' + 24 (^/« - 26 cd^P^ 36 d«cy + 24 - 2 oi»«» - 16 a} h'ep + 5 de"/ - 35 cd^f- 42 rfe* - 10 -38 d^e + 12 6'crf/« - 4 «< + 42 ce* + 24 6V<5/« + 8 + 34 a> hHp + 7 ct^f + 35 b'c^dp + 2 cP^ + 56 (?d^P^ 2 - 9 ey - 12 rf*?/" - 26 cV/+ 26 cP«» - 34 c'cfey- 52 ! + 5 6V/« + 6 efo« - 22 cgPc/" + 72 6V^« + 10 cV + 28 + 2 cdef + 42 6'cy + 10 cde" - 124 c»cy- 54 cd?ef ^ 52 j - 12 C6^ • • • c'def- 72 rfy + 13 c^d^ef+ 64 ccPe^ - 32 -24 e^f + 54 cV - 106 d"^ + 26 c^d^ + 46 d^f - 18 + 52 (f«e« - 91 ccP/ + 76 6^cy* + 9 cd'f - 37 rfV + 12 1 6Vc/ - 68 ccPc* + 210 cW«/- 76 cd^e^ - 50 a« 6V' + 1 ' -22 c»rfy_ 64 d^'e - 99 cV - 56 cPe + 21 ei^/« - 13 i -52 c«cfe« + 14 ¥c*ef - 13 c«c^/+ 10 a« 6y» + 2 ey + 12 + 34 ccPc + 204 i?d^f- 10 c^cTe^ + 296 6»c^/» - 32 b^c^eP - 2 + 8 ^ - 93 i^de" + 128 cd'e -260 c^y^ + 24 ccP/- + 38 - 1 V'i^df + 37 c'c^e - 184 c^» + 72 d^f • • ■ crf«y- 7 1 + 18 cV + 86 cc? + 72 a' b'ep - 17 e* • • • ce* - 30 -25 c*(^c - 208 a« 6*c(/^ + 4 6»c4/'« + 40 6V(y« + 16 d^ef - 34 + 10 c«rf^ + 86 ^/ - 42 cey + 22 c^^y + 91 d^^ +35 - 2 a« 6^c/« - 5 6V/« - 8 d"^ + 106 cd^ef- 14 6Ve(/^ - 34 + 10 cfe/ - 12 cd«/* + 124 (ie^ -105 cd^ - 105 c»ey + 22 -28 c • • • cc* + 105 6V/« - 24 d^f - 86 c«rf«c/- 8 + 30 6Ve/ + 34 rfy + 56 c«(fe/-210 cPc« + 110 i?df? + 50 + 32 crfy - 46 C^6» - 130 c»c» + 130 6V/« - 12 ccfy + 25 -35 cd^ + 105 6V«/ - 26 cd"/ - 128 i^d^f- 204 cdJ'^ - 70 -50 c^e - 20 <?d^f- 296 cd"^ + 170 i?^ + 20 rf*c + 15 1 + 30 l^i?df + 50 i?d^ - 170 rf^e - 25 (^d^f + 208 6V/« + 9 - 12 c»c« -110 ccPe + 340 b'c*e/ + 99 c«c^e« + 170 c*<fe/+ 1 + 70 c*ci»e - 170 c/» - 60 c»rfy + 184 cc?*c - 250 cV -30 -40 cd?' + 115 h'i^df + 260 c»c£e« - 340 ^ + 60 c'ciy - 10 -15 6V/ - 21 cV + 25 c'd/'e + 150 6Ve/ + 93 c'(^c« + 40 + 10 d'dA + 250 c'cPe - 150 ccf - 40 c'd^f- 86 <?d^e - 10 > • a c«rf» - 150 c«c^ « • ■ b'c'df - 72 ^d^ - 115 C€^ 6V« - 60 6«cy - 72 c»c« + 60 c^cPc + 150 &d^ + 40 c"cfe + 40 c*d^e cW - 40 • c^d^ « • • c'd* ± 3 =k= 6 Jt^ 6 ^ 1 67 99 70 27 ± 24 ± 6 136 536 536 577 266 134 182 594 954 961 « 944 248 5«, uf ^ 388 dbl234 :tl566 rtl566 ±1234 db38a 288 TABLES OF THE CO VARIANTS M TO W OF THE BIKABT QUINTIC. [l43 Q. No. 25. Q'. No. 26. Q. No. 25. Q'. No. 26. a« bof* • • • + 1 a» 6V + 27 - 3375 a* b'ef* • • • 20 h^t^dp - 48 -1- 5760 b'cd/* + 1 - 120 <?i?f + 3 - 600 c<?P - 1 + 160 cd^ef + 106 - 16000 (Pep - 3 + 360 ed^ - 81 + 9000 d^f + 5 - 640 d^f - 38 + 6400 «• - 2 + 256 rf»c« + 38 - 4000 a* b'df* - 1 + 160 6V/« + 18 - 2160 «•/* + 1 10 i^def - 30 + 7200 W/» - 3 + 360 &^ + 38 - 4000 cdeP + 11 - 1640 t^d^f + 8 - 3200 <^f - 5 + 320 <?d^^ + 25 + 2000 d»/« + 12 - 1440 cd^e - 57 • • • c?^f - 30 + 4080 rf» + 18 <fe« + 15 - 1920 b'i^ef - 9 6V«/» + 12 - 1440 c*d*/ + 6 <?cPp - 21 + 2640 C*(fo« - 57 <?d<?f - 34 + 4480 c'd'e + 74 <?^ + 22 - 2560 c'd* - 24 c^rf + 78 - 10080 y'cHf • • • edP^ - 48 + 5760 c«c« + 18 ^f - 27 + 3456 c^cf'c - 24 <?«» + 18 - 2160 c*rf* + 8 a' VcP + 5 - 640 d<P - 5 + 320 «y . • • - 180 l^<?ep - 30 + 4080 eePP - 34 + 4480 edff + 133 - 14920 ct^ - 54 + 7200 «P«/ - 18 + 960 dP^ + 3 ~ 600 b^i*dp + 78 - 10080 e^f - 18 + 960 <?d^ef - 220 + 28480 <»d<? + 106 - 16000 cd>f + 93 - 11520 c<i*«s* - 30 + 7200 Thesm 018 for Q' are d*e - 9 • * . 1 = 1 b'>(*P c*def - 27 + 93 + 3456 - 11520 776- 21266- 68656- - 780= -4 - 21250= +6 - 68660= -4 «!*«• - 38 + 6400 87816- - 87815= +1 <?d^f c'rf's* — 4.9 + 5120 - 3200 + 8 128505- -128505= c'd'e + 6 • • • cd" . • • • • • a* 6»/» - 2 + 256 6V^ + 15 + 1920 «P/» + 22 - 2560 d^f - 54 + 7200 ± 6 til 128505 169 525 424 ^1124 3] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 289 R. No. 92. • • • a» Vc^f - 15 a* 6y* • • • a^ 6V • • • o» jy • » • a« 6»d»e/ + 2 • a • rf«e/- 38 a' 6 V* • • • 6Vc^ + 18 Veep • • . dJ'd' + 15 • • • cfe* + 46 h'cdp • • • i^^f - 66 dy« + 1 6Vrf/«- 32 - 1 6»cy« + 3 c^P • • • cd^tf ■\- 20 d4?P- 2 c»e»/ - 39 + 6 c«(fe/+ 102 (^e/« + 2 cd^ • • • «y + 1 cWe/- 24 - 4 c»«» - 16 cfey - 4 d*f + 58 a* 6V' • • • c«dfe» + 175 - 3 cdy+ 76 6» + 2 d'e' - 50 l^cdp - 6 c^/ + 25 + 1 crfV - 175 a» }^dp • • • 6V/' - 6 c^P + 6 cd»c« - 120 + 1 d^e + 35 ^P • • • c^def -h 72 cPeP + 3 c^e + 15 + 2 6*cV - 42 h^i?P - 2 cV + 50 d4?f • • • 6V/« + 9 - 6 c»(^/~ 182 cdeP • • • c^d^f- 156 «• - 3 d'def + 106 + 4 i?d^ + 120 ct^f + 4 c«rf*e» • • • 6V/' + 3 c*e» - 35 - 3 c>d»« + 150 d»/» - 14 cd^e + 90 <?deP- 3 c»dy- 60 + 3 c(i» - 70 d^^f + 30 d» - 30 <?«*/- 6 c»d«e»-150 - 18 h^&df + 126 d^ - 18 6V^/ - 24 cePp • • • c»rf*c + 176 + 17 &^ - 15 6Vc/« + 14 cW + 94 cd^^f-h 3 crf« - 45 + 22 c*d»« - 175 cW/« • • • i^d^ - 90 c<fo* + 6 6Vc/ - 36 - 21 (?d* 4- 75 (^d^f- 66 c»^e • • • #«/ • • • d'd^f^ 21 • • • 6V/ - 27 (?^ + 26 c«c?» + 10 <?«• - 3 c»rf6> + 70 + 13 c«(fe + 45 cc^ef + 56 h^d'df - 18 a' 6'd/» + 4 c*d»« - 75 - 12 c»(^ - 20 cd^^ - 18 C«6» + 30 «•/' - 4 c>d" + 20 - 21 ^f - 18 <^d^e ~ 10 6V/» - 1 - 3 d^^ + 6 c^c^^ • • • c<fo/^ + 18 + 32 a» 6»c/« + 4 w*/ - 16 - 9 dep - 4 ePP - 13 - 1 ^/ • • • cP^/- 3 • • • 6 V«/« - 30 d^ + 15 + 6 cd^P^- 66 h^<?fp - 22 + 16 cde^f • • • c'cPP* 12 - 18 c^ - 18 (?d4?f+ 18 - 3 d^ef - 84 c*e* + 38 + 3 d^iS" + 66 C€Pef + 32 - 18 h'i^dP-- 56 cdP^ - 102 + U c^^f ^ 84 <Pf - 18 - 41 ^d^tf % » * d**" + 42 + 39 c^d^ - 20 Ve^dp + 3 • • • cd^f + 40 cV/ + 41 - 32 cd^^ - 72 <?cPef- 84 - 2 (Pe + 24 <?d^ - 76 + 84 6V/« + 18 c'd'/ + 33 + 24 c*de/- 40 «•<?«• + 182 -106 c^e» - 58 cd»« - 126 + 36 c'dy • • • rf' + 27 + 18 c'cPe' + 156 a'iV* - 1 ~ 33 ± 8 c«c^e - 94 ± 4 deP - 17 :&: 2 - 25 98 cc?* + 18 136 «y + 18 21 + 60 300 a' b'P - 2 476 6Ve/» + 21 465 - 21 780 6*c«/^ + 18 478 cd>/« + 21 693 + 3 ±1181 d'P - 26 :tl094 cd^f- 14 :lrll81 - 6 d^f + 18 ce* — 45 \«^ y)'- C. II. 37 290 TABLES OF THE C0VABIANT8 M TO W OF THE BINABY QUINTIC. [l43 S. No. 93 6m ; S'. No. 93. (•5«. y)». CJoof. *• S • 3' Coef.x» S 8' Coef. j:*// s 8' Coef. £*y 8 V a* by* • • • a} l^d'ef - 66 + 528 a* 6V* + 9 a> b't?^ + 6& Vcef* + 9 d^e" + 72 45 deP — 45 C(Pef + 7B\ cpr + 21 h^(?dp - 21 - 2592 ^P + 36 c^^ - 1&& d^r — 78 C»6>/ - 96 - 9747 a» 6y* — 9 d^f c*/ + 48 ed^ef + 36 - 8496 6W* — 18 d*«« V cfVeP — 9 ^d^ + 213 + 26610 d^P + 243 b'd'dp + L ^*- h^cdp — 162 cd^f + 120 + 8544 d^P + 9 cV/ __ ^^ c<?P + 99 C(P^ - 303 - 16650 eV — 216 <?d^tf — dPtP + 309 d^e + 51 + 720 V'&dp - 3 — 351 C»d8» — A*/ + 12 h'^P + 9 + 972 i^^P + 3 + 144 c*d^f — «• — 240 ^def + 174 + 24624 cdPeP + 24 + 1836 i^d^i? + 1 - 6V/» — 2 — 81 cV - 36 - 5040 cd^f - 42 — 2592 cd^e + <?der + 15 + 1026 i^d^f - 204 - 15984 C6» + 18 + 1152 d' — <!»«•/ — 9 — 768 c»e?«» - 174 - 29340 d^P - 18 — 1458 W/« — eeP/* — 9 — 738 (^d^e + 330 + 34320 (Pt?f + 33 + 2268 <^def + «?«»/ — 6 — 664 cd^ - 99 - 8640 (i»«* - 15 — 1008 c««» + ede* + 9 + 1056 6V(5/ - 63 - 7776 a^h*ip • • • + 63 <M^f + d*ef + 9 + 756 ^d^f + 66 + 6184 b'cdp + 6 — 234 cV?«« • <?«» — 7 — 696 <^d4^ + 99 + 12960 cep - 6 — 18 <?d^ + o?Vdp • • • + 120 d^e - 147 - 14400 d*ep - 24 — 3231 c'd' + «•/• • • • — 21 c>d» + 45 + 3840 ^f + 42 + 4293 a* ft»c/» + _ v<»p + 6 + 486 a* 6y • + 2 + 192 ^ - 18 — 972 deP — ^^ edef* — 30 — 2160 h^ceP - 15 - 1440 h^i?P + 3 + 810 ^f + tfS c^f + 18 + 1023 d^r - 6 ^ 192 <^dep - 78 — 3826 h^eP .^ ^ d}/* + 9 + 120 *y - 18 - 1080 <?i?f + 69 + 4032 cdy* + Sf d^f + 6 — 1053 <5* + 27 + 2025 cd^P + 93 + 7938 cd^f - 9^ (fe« — 9 + 1314 b'c'tif* + 24 + 1728 cd^^f - 61 — . 9360 c«* - zr' h^eP — 15 — 1863 i?^f + 61 + 4410 ede' - 33 — 864 d^ef ^r 60* <?€Pr + 21 + 2538 cd^ef + 102 + 5280 d>ef - 57 — 1296 (?«» - 45 <?def — 6 + 2340 cd^ - 171 - 13500 (Pe" + 54 + 2700 h^<^dp - 39V <^e* + 18 + 672 ^f + 6 - 4800 h\^eP + 24 — 324 t^^f + 45 '^ ed^ef + 30 + 2820 d^i? + 18 + 7800 i?d^P - 36 — 2484 i^d^tf - 108 «?«• — 61 — 7812 l^c'P - 9 - 648 i^d^f - 9 + 6624 c>cfe» + 96 r- dy — 36 — 3024 &def - 210 - 14040 c»«* - 64 — 6912 cdyf - Ill ^ d*** + 39 + 4672 <*^ + 43 + 3076 i^d^tf + 24 — 4428 edP^ + 147 — JVt^' — 3 — 324 <?d^f - 120 + 9120 c>d»«> + 129 + 12672 d^t - 30 + c'e'/ + 46 + 3888 C>(?«' + 345 + 16350 cd^f + 9 + 1944 6V/« + 9 + i^d'ef — 84 — 8748 cd^t - 87 - 19200 cc?*e' - 114 — 9072 cW + 6 — c'de* — 63 — 4800 d" - 2 + 4800 d^e + 27 + 1944 <?*«> - 48 — ^d^f + 45 + 4248 IMef + 72 + 4860 a^h^dp - 3 + 144 i^d^f + 234 - <?dPt? + 150 + 14620 d'd^f + 240 - 3240 ^P + 3 — 243 i?d^^ - 150 -» (xPe — 117 — 11448 (H^ - 192 - 8100 b'i^P - 6 — 900 C\^J - 108 - cP + 27 + 2692 (*d^e - 186 + 9000 edeP + 108 + 10620 erf* + 57 a> b*e/* — 6 — 576 c»d» + 96 - 2400 c/f - 96 — 8586 h^^tf + 9 def* + 15 + 672 h^d'df - 144 d^P ~ 21 — 864 i?d^f - 141 «•/ — 9 — 459 C«6> + 18 d^^f - 48 — 1215 &d^ + 87 V<?eP + 30 + 3466 i^(Pe + 201 df^ + 63 + 1215 e^e + 96 cd^r — 15 — 864 c'd^ - 87 6V^ - 24 — 1836 i^d" - 51 ed^f + 24 + 2094 h'<?f + 27 (?d^P - 123 — 16812 b^e'd/ + 27 C9* 45 3915 i?dt i^d^ - 46 + 20 cW/ + 147 + 6651 - 18 - 21 + 12 For the Namerical Verifications for S see further pp. 304, 806. 78 db as 3258 414 41253 1284 124524 1292 68640 db8028 :k 287758 ± 78 480 927 96 6 db245i 143] TABLES OF THE OOVARIANTS M TO W OF THE BINARY QITINTia 291 S. No. 93 Us; S'. No. 93. f« s 8' Coef. xy' S 8' Ck>ef. y* S S' Coef. y» S 8' • • • _ 9 a^b'i?^ _^ 60 + 4320 a* 6 V* • • a a^ b^d'd - 72 - 4860 1 • • • + 9 i^ef + 36 + 14544 a» b'd/' — 9 b^'deP + 36 + 3024 • • + 45 i^dPi^ + 108 — 3060 ^P + 9 dd^P ~ 45 - 4248 t • • • + 18 c^f — 24 — 5184 ¥df* — 21 dddf - 120 - 8544 1 • • • — 63 cd"^ — 6 + 1620 cdep + 162 dd - 6 + 4800 •B • • • — 243 d'e — 9 • • • cdp — 120 dd^ef + 204 + 15984 r» + 3 + 351 h^<*dp — 9 — 1944 rfy» + 2 + 81 d(Pd + 120 - 9120 /• - 6 + 234 i^i^f — 51 + 3888 d}dp - 6 — 486 dd^f - 66 - 5184 ■ + 3 — 144 d'd^ef + 96 — 1296 de*f + 6 + 576 dd'd - 240 + 3240* 1 - 3 — 810 C*d8» + 111 — 1440 d - 2 — 192 cd^e + 144 • • • f + 6 + 900 f?df — 27 + 576 a« h'cP • • • + 78 <f» - 27 • • • - 3 ■— 288 c»d»«« — 234 + 360 dtp • • • — 99 d' b^ep • • • + 240 • • • — 36 c«d»« + 141 • • • ^P • • • + 21 b^cdP - 9 - 1056 1 • • • — 9 erf' — 27 • • • h^deP • • • — 309 cdp + 9 - 1314 1 - 3 — 144 a^l^dp — 18 — 1152 cd^P - 15 — 1026 d^eP - 18 - 672 ri + 6 + 18 ^P + 18 + 972 cddp + 30 + 2160 ddf + 45 + 3915 - 3 + 243 6V/« + 15 + 1008 cdf - 15 — 672 d - 27 - 2025 1 - 24 1836 cdep + 33 + 864 d'ep + 15 + 1863 l^dp + 7 + 696 n + 24 + 3231 c^f — 63 — 1215 ^df - 30 — 3456 ddeP + 51 + 7812 r + 78 + 3825 (PP + 54 + 6912 dd + 15 + 1440 ddf - 72 + 45 f - 108 — 10620 d^^f — 66 — 12960 V'ddp + 9 + 738 cd^P + 63 + 4800 + 30 + 3888 de' + 27 + 6075 ddp - 9 — 120 cd^df - 213 - 26610 1 - 24 + 324 6V«/» — 54 — 2700 dd'eP - 21 — 2538 cdd + 171 + 13500 f + 24 + 1836 i»d^P — 129 — 12672 dddf + 15 + 864 d^ef + 36 + 5040 • • • — 756 C'd4?f + 186 + 18900 dd + 6 + 192 (Pd ~ 43 - 3075 1 + 18 + 1458 c>«* + 45 — 6075 cd^p + 3 -4- 324 l^deP - 39 - 4572 /' - 93 — 7938 cd^ef + 54 + 12960 cd'df + 21 + 2592 dd'p - 150 - 14520 f + 21 + 864 cd^(^ — 96 — 10125 cd^d - 24 — 1728 dddf + 303 + 16650 r + 36 + 2484 d^f ~ 54 — 5760 d^ef - 9 -^ 972 dd - 18 - 7800 «y + 123 + 16812 rf*e» + 48 + 4500 d^d + 9 + 648 dd^ef + 174 + 29340 4 - 51 — 7488 l^^P + 114 + 9072 a' b*P • • • — 48 dd*d - 345 - 16350 f - Ill — 15228 c*«y + 9 — 2970 b^ceP • • • — 12 c^f - 99 - 12960 t + 39 + 7128 f?d^ef — 150 22500 d'P + 9 + 768 cd'd + 192 + 8100 + 27 + .3888 dd^ — 147 + 13950 ddP - 18 — 1023 d^e - 18 • • • - 9 — 1944 dd^f + 93 + 9360 ey + 9 + 459 b'ddp + 117 + 11448 • • • + 216 d^d + 150 — 6300 l^ddp + 6 + 564 ddf - 51 - 720 1 + 42 + 2592 ed^e — 87 • • • ddp - 6 + 1053 dd'ef - 330 - 34320 •a - 42 — 4293 d? + 18 • • • cd^eP + 6 — 2340 ddd + 87 + 19200 1 - 69 — 4032 b'dp — 27 — 1944 cddf - 24 — 2094 dd^f + 147 + 14400 • + 96 + 8586 ddef — 30 + 6480 cd + 18 + 1080 dd/'d + 186 - 9000 - 27 — 3645 dd + 30 — 3600 d>p - 45 — 3888 dd^e - 201 • • • - 33 — 2268 dd^f — 6 — 2880 d^df + 96 + 9747 ccT + 45 • • • /^ + 51 + 9360 d^d + 108 + 1800 d'd - 51 — 4410 b'dp - 27 - 2592 r + 48 + 1215 dd^e — 96 • • • b'dp - 9 — 756 ddef + 99 + 8640 rt + 9 — 6624 dc? + 21 ddep - 30 — 2820 dd + 2 - 4800 y - 147 — 6651 b^def + 27 • ddf + 66 — 528 dd^f - 45 - 3840 + 39 + 4050 dcPf — 9 ddJ'P + 84 + 8748 dd^d - 96 + 2400 • + 78 + 4968 ddd — 57 dd^df - 36 + 8496 dd^e + 87 • • • - 45 — 2970 dd^e + 51 ddd - 102 — 5280 dd^ - 20 • • • •1 + 57 + 1296 d(P — 12 cd^ef - 174 — 24624 /• - 24 + 4428 • cd^d + 210 + 14040 y - 78 — 18612 d^f + 63 + 7776 He 9 =k 12 1548 426 45999 912 62019 1101 92853 =b 8 db 828 123 10920 1071 79779 1821 146226 :fe2451 ±202428 ±8023 ±237753 37—2 292 TABLES OF THB OOVABIAMTS M TO W OF THE BINAirr QUINTIC. [14 ( T. No. 94. X coe£Scient a coefficient a* VcP a'lt'^P _ 20 a> ft«c»d'«/» + 153 a« 6V/« - 6 def* d'^P + 33 <?d^f — 390 c»cfe/« + 240 «•/' <foy — 48 cV — 234 «•</■ + 179 a* bV «• + 27 ed*P — 114 c»d»/« - 144 b'cef* V&fp + 39 cd*<?f — 308 c'cPe'/ + 306 <P/* <?d}p — 105 c«P«* + 735 c'd^ - 765 de>P <?d^P + 18 d^ef + 208 cct^f + 28 e*/* <?t^f — 6 d*e» —- 283 cd"«» + 280 VtMf* - "i ed?eP + 114 wp + 27 <p/ - 88 <?^P + 1 edPtff ~ 67 <*deP — 396 (PeF + 40 ccPef* + 7 edtf + 12 c'e'/ — 337 6"c»«/« - 63 cd4?p - 12 dPp — 6 <?d^P + 222 c\iPP + 42 c^f + 5 d'e'/ + 3 <^<P^/ + 783 <?*cfe>/ - 798 #/> - 6 d*»^ — 12 c«<fo« + 880 cV + 176 cP^P + 12 l^d'dP + 90 c'd^ef + 93 d^iPef - 224 dfi^f - 7 <!««•/» — 198 e'd'e* — 1986 c»rf»«» + 1366 d^ + 1 (?d}eP — 9 cdff — 240 cW/ + 368 c? i'e/" • • • <?d^f + 238 cd>^ + 1098 c»d*«« - 1026 Vcdp + 2 <!»«• + 116 d'e —- 144 cd^e + 60 c^P - 2 ^dfp — 6 b'ifcP + 81 d» + 30 6?tP - 7 i?d*^f + 108 <fd'P — 54 l^^dp • « • d<?P + 12 c**?** — 613 (fde'f + 570 if^f + 252 «•/ - 6 cdPef — 294 c»«« — 148 c»c?«/ + 798 6'cy« + 3 edV + 513 e*(Pe/ — 1116 C»d8» - 700 <?deP - 30 d'/ + 108 c*d'e* — 627 cV/ - 578 i?^P + 21 (?«• — 153 c«d»/ + 474 iN^i? - 370 ecPP + 44 bfd'p — 27 <?d>^ + 1662 c'cPtf + 880 edP«?P - 69 <?deP + 108 <?d^e — 1185 <NP - 240 «fey + 62 c»«'/ + 194 eeP + 243 6V/« ... c«« - 28 «*</•/* — 42 iVdp • • • <?def - 486 d^eP - 6 c**?**/ — 663 c'ey — 216 <?^ + 60 d»e'/ - 8 <*dif^ — 274 (fd'f/ + 369 «•(/»/ + 312 <?«• + 11 ed^tf + 570 d'd^ + 340 c«dV + 645 6Ve/» - 6 €»<?«• + 914 (I'd*/ — 149 (*d^e - 735 ifd^P - 11 <?dff — 163 (!»(?«" — 730 e^ + 190 <?d^P + 96 cW«» — 1032 C*(P« + 488 6V«/ + 81 <?eV - 64 ccPe + 486 e'd' — 102 C»(i»/ - 54 <?df«P - 66 d* — 81 a' VP — 2 c»cfe» - 136 <?(P<?f - 29 a» VeP + 7 Veep + 20 (?d^e + 150 <?dtf + 68 deP — 16 d^P — 24 c»d» - 40 ecPp + 18 ^P + 9 d^P + 72 ed^f + 76 b\?eP — 53 • «y — 64 cd^(^ - 78 cd?P + 104 v<?dp + 16 ^ 26 d^ef - 27 cd<?P — 150 c"«y — 129 486 a* b*dp + 24 - 1 d}iP + 117 48 cd'ef* ed^f + 108 72 8788 9116 6880 v<*p + 1 d^ff df + 138 €«■ + 135 1 ^ - 8 1 108 d*P + 84 :!: 90196 edeP + 46 v&dp — 82 d'e'/ — 112 and see farther p. 806. c^P - 30 c»«y + 316 d»«« • • • 143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 293 T. No. 94. y coefficient. y coefficient. t^b'df' a« 6"crf»ey« - 18 a' b^d'dp - 75 a<>6^jd»ey - 880 </•* cd^f + 150 <^^P - 3 ccPtf* + 765 a* hhp c«« - 72 &d^eP - 108 d^ef + 148 def^ dl'ep + 198 <?df?f + 308 ^•«' - 175 ey d»ey - 315 c»«» + 112 6V/» - 24 h^(?ef^ rf««» + 129 <?d^P + 663 d'dep - 513 cd^f + 1 h'd'eP + 6 c«rf»ey - 783 cV/ + 283 cd^P — 2 c8rf»/» + 66 • c»rf«e* - 306 i^d^P ~ 914 c^P + 1 <*d^p - 114 ccJ'e/' - 570 c»rf»ey + 1986 ^fp — 3 c>«y + 48 cd^^ + 798 <*d^ - 280 d^^P + 8 c^d^ep + 9 ^/ + 216 t^d'ef + 527 d^f — 7 c«rf»ey - 153 (i»«» 252 c«rf»e» - 1365 ^ + 2 (?d^ + 108 6^cy» + 27 cd^f - 340 (^¥p • • • cd^P - 108 C'flfe/^ + 294 cd^^ + 700 b'cep • • • crf»cy + 396 c»ey - 208 cTe - 60 d^P — 1 cd^^ ~ 240 c'd^^f - 93 6»c*e/« + 153 d^P + 2 d:'ef - 81 cV/> - 570 ^d^P + 1032 ^r — 1 d"«» + 63 c^cfo* - 28 (?d^f - 1098 h^i?dp — 7 6V(^» - 18 c*c?'e/' + 1116 c»«* - 40 (?^p + 7 c»cy> + 6 c'c^c* + 224 i^d^ef - 1662 cd'ep + 30 c^d^ep + 6 c»dy - 369 c^d^^ + 1025 cd^P — 46 d'd^f + 114 c«d»c« - 798 c'rfy + 730 c^f + 16 c*«» - 84 cd?e + 486 c»(^e» + 370 d^P + 6 <?d^p + 42 d» 81 c^cJ'e — 645 (Pe'P — 39 <?d^(?f - 222 6Vfl/« - 108 c^ + 135 iP^f + 53 c>rf»«* + 144 i^d^P + 153 6^c'(y» - 486 A» — 20 (?d^ef + 54 c«rfey + 240 cV/ + 144 h^p + 6 c"rf»«» - 42 c««* + 88 d^d^ef + 1185 edrf* — 44 ccPf <^d?ef 474 c«(ie» - 60 e^p + 20 cd^^ i^d^^ - 368 dd^f - 488 <?d^p + 11 d^e d'cPf + 149 dd^d - 880 c'd'^P + 105 a* mp - 5 c^d^^ + 578 dd^e + 735 c'd^^/ — 104 ey* + 5 c'cJ'e — 312 dd^ - 150 ^^ + 24 b'i^P + 7 0*6? + 54 6V/« + 81 I cd^eP — 90 edep - 62 a« 6 V* 1 ddef - 243 cd^^f + 82 c(^P + 48 cfe/» + 28 dd 30 C(?<J» — 16 (PP + 64 ^P - 27 dd^f + 102 d^p + 27 d'e'P + 6 fc»C«(5/» - 11 dd^d + 240 d^^f — 27 efey - 117 cd'P - 68 c'c?*e — 190 d^^ + 6 «• +54 c(iey« - 12 c'c?' + 40 a* 6V* • • • • 6Ve/» + 8 C€*/ + 108 l^cdp + 12 c'd'P + 29 d^eP - 116 e^P — 12 i^d^P + 57 d'e^f + 234 i 12 d}tp — 21 c>«y - 138 cfo» 135 895 d^P + 30 cfPeP - 238 6Vci/^ + 78 1650 // — 9 ceP«»/ + 390 c»6y + 12 6511 4 *« £^£^£\ ^<?P + 12 69 cd^ - 72 rfy« - 194 i^d^eP + 513 735 11628 i?dep =1=20196 f?^P — 33 c^«y + 337 c»«» ... and Bee farther p. 806. \ ed^p — 96 d»^ - 179 cc^/« + 274 5*. y)' 294 TAPLW OF THB C0VABIANT8 M TO W OF THK BINABT QUINTIC. [14 U. No. 29. a* b"/* . a* J'ePa* - 22 a« 6V«i»/» - 108 a'Wi^P - 90 o» VdPP - a* 6 V» 6V/« - 4 cW/- 42 AP^ - 42 dV/ - 6»«i^' <!*</«/» + 36 <!»«?«« + 298 <W/ + 674 d»«* + «!«•/* ^^P - 16 odV" + 242 <!♦«• 4 b*<*dp + tPef* (?tPP - 22 «iV - 294 c»d'/» + 394 <ft?P + d^f* ifd^P- 50 d»/ - 72 c»dV - 662 <?d^P + *P ifd^f + 16 «?«• + 78 <?<P^/- 714 (?dif/ - t^Vf* <!»«• + 16 V'^dp - 6 <?cPnf - 498 <!»«• + Vdf* c»«?e/» + 54 *•«•/» + 62 c'dV + 1246 <*d^P + ^f* e^iff + 46 c'dV - 108 ed'/ + 224 <?^^f - jvy* <!•«?«• - 60 fl»«fc»/ - 164 od»«« - 516 <?^P^ + cdef* t^P - 6 c»«» - 24 d"* + 48 afef + tPP ccfeV - 70 cy/* + 63 6V/» + 18 cdV - <?«•/» «#«♦ + 56 cV««/ + 394 i'ds/' + 242 dPf + d^P rf'e/ + 18 cy»«« + 194 cV/ - 128 <P(j» + «•/ «<•«• - 14 edftf - 324 o'dy - 324 6»c»/» - 6V«/« .. a» ft*/* « • a c»dV - 440 c'dV/ - 498 (*d^ ~ c«<?/« - 1 6««/« • • • c'd'/ + 78 c»d(J* + 136 &^f + c'de»/» + 2 «?/« - 1 «»<?«• + 428 ««V" + 1078 e<d»/« - c»«y» - 1 Ay* + 2 «*•« - 180 (A?** + 206 cW/ + «?«/» + 6 «y» - 1 6P» + 27 c»dy - 342 c«d«» - cJP^P - 1 6 W4^« - 16 a> 6V* c'd»(j» - 804 c'dV + «W/ + 1 4 «V/» + 16 ft»«(^« + 14 (Me + 506 c'd»e« - ce' 4 tdPeP + 82 e^P - 14 cd» 90 cV/ - <?/• - 4 cefe»/« - 132 <P^ - 32 6V«/-» - 72 c^V + (^e*/* + 1 1 c^f + 50 d«»/» + 50 e'tPP + 78 eJPe cP*?*/ - 1 ««•/• - 16 «•/ - 18 <?d^/ + 224 d» <;■«• + 3 df^P - 14 bVp - 10 <?<^ + 16 6«c'5r + o» 6»^» dV/ + 60 «W* - 30 (?tP^ - 342 c»d'/» + de/* d* - 30 c»«»/» + 60 (fdf* - 220 C«d9»/ - «•/• 6V/« + 11 edVp- 48 <fd>f + 106 c»«« + 6V«/« <>»&/• - 30 c«iy + 16 <fd>^ + 392 eepif - cd»/' + 2 c»«y« - 14 ed^/ + 88 cV« - 222 c»dV + «W/» - 4 cW/' - 60 <!<• - 86 <?6P + 40 cW/ + c«y» + 2 <!»<?«•/• + 168 d*ep + 112 a« Vdp - 4 c«V + d'«/» - 6 «!**•/ - 48 d**"/ - 204 t?P + 4 c»d«« - ^^p + 1 6 c»<C - 4 (Pif + 102 6V/« + 3 cV + rf**/ - 1 4 cdV* - 48 6»c«</^ + 50 edrf* + 24 ^ + 4 «?«•/ - 2 c»d»/» + 46 <?d?p - 2 e»««/ - 204 ©«•/» - SO 6'c"<(/*« + 6 «P«» + 6 d»/» + 16 ed}«f + «!•«•/» - 6 #/» + 62 d»e»/' - 4 c'de» - «!»</•«/» - 5 rf-e*/ - 90 <?(P^ - 170 d»«/ - 36 efd*/ - c'd8'/»+ 8 2 d»«* + 39 «!**• + 308 if + 27 c«d»«» - oV/ - 3 2 iVe/* - 28 cV**/ + 42 i'cdV* - 104 c*d»« + cd>P + 3 6 c<d»/» + 64 «?/> - 164 c»^ - 22 cW - «?«•/• - 3 <*d^p - 48 cdV/ + 674 c»d»/» - 60 6V«/» + crf»««/ - 3 cV/ + 112 etP^ - 590 <fdt?p + 6 «•*/ - e(i«* +2 4 c»d'^ + 82 dV" - 128 cV/ + 102 c»e» dftP - 2 .8 c»d»«'/ - 170 (i>«« + 138 cdW/ + 308 «^d»/ + «W/ + 5 <*d<f - 104 W4f* - 70 cde* - 234 c'dV + (Cd* + rkSe, ^464, db2608, :k7378, d:6878, together :&: 17264: and see farther p. 807. 143] TABLBS OF THE COVABIANTS M TO W OF THfi BINAftY OUtNtlC. iu V. No. 95. (•$«, y)». » ooeffioient. 0^ 6»c/* <f b*t^' ^ 2 a» 6*orf»«' + 876 a« 6Vrf"<^ + 2800 d^ ^f* + 2 d^f + 162 i^d^rf + 6624 «•/* V(?P — 16 rf»fl» — 162 i^d^^ + 2052 «• 6y« cdeP + 32 a« }^rf^ + 14 c^f — 918 6'c«/» c^P • • • def^ — 6 ccTe* — 2304 (Pf tPp — 8 «•/• — 8 d»« + 486 «fay* «?«•/» + 80 b'c'ef' — 50 c'dp • • • «•/* d^P — 160 ed"/' + 90 h'(?i?P + 504 V<?df* — 2 </• + 72 cd^P — 120 i^d^p — 576 <(•«»/< + 2 ve«p + 84 c^r + 60 i^d^f — 2288 «?«/* + 10 ^dfp — 104 ^rf^ — 280 i^4^ + 1172 ed^P — 16 ifd^P — 160 d^^P + 300 <*d*p — 124 ofP + 6 «;•««/• + 60 d^f + 216 c»rf»«y + 4336 itf* — 6 cefep + 320 ^ — 216 dd^f^ — 2540 ^^f* + 12 et^P + 80 h'i^df^ — 160 d'd^ef — 1912 dV/» — 10 odff — 496 &^P — 80 c*d*«» + 2100 d^f + 6 eg + 252 ^d^P + 1280 ^d?f + 240 «• — 2 dPp — 72 <?d^P • a • &d^^ — 1560 a* 4«^ • • • dVp — 420 c»</" — 312 &d^€ + 810 Vedf* + 4 ePe*/ + 860 cd^P — 440 ecP^ — 162 c^f* — 4 dV — 404 cd^e^P — • 2160 a» 6y» — 4 d^ef* — 10 b'^dp + 96 edS^f + 1740 h^CBP -1- 22 «&•/» + 16 cV/' — 120 cd^ — 216 d'P ~ 26 «•/• — 6 i?d?^P — 560 d^tp + 2344 dd'P + 76 WP + 6 «?dgp + 160 d^ef 3240 ^P • • • ^def* — 26 «»«•/ + 304 rf»«» + 1244 }^i?dp + 124 «•«•/• + 8 <?d^P + 280 WP + 72 f^^P + 368 ed^f* + 32 ^dfgp + 1440 (MeP — 240 cdFep — 688 «?«•/» — 116 <?<Pe*f — 900 &i?P + 940 cd^P — 192 tdi^P + 180 e'dtf - 376 f?d^P • • « ^f • • • «*•/ — 78 ed^eP — 1296 &d^^P — 1320 d^P + 400 <ft./» + 24 ed'g/ + 80 &df^f — 2640 d^^P + 984 <?«!•/» — 20 cd^g + 832 fl»^ + 908 d^^f — 2160 ««•«•/ — 44 «r/» + 432 i^dSp + 600 d^ + 1080 dg + 34 dfgf — 72 &d^^f + 3360 h^d'p — 60 6»«V* — 30 dP^ — 240 (?d^f^ — 168 ^dep — 480 ^d^f* + 4 Wp — 36 ed^P — 1656 &^p — 1580 ifd^fp + 240 ifdtp + 288 fid^^f + 3408 i?d?P + 40 «•«*/» — 130 (fgp — 56 ed"^ — 3480 f^d^^P + 2040 ^^»P — 160 <^d}p — 140 d?ef — 1008 i^dt^f + 2910 (?(Pt?P — 280 (fdP^P — 480 d^i^ + 1224 c»^ — 810 ^d^f + 332 iNitff + 420 h'ifep — 144 6d^€f^ — 3420 «»«' — 54 «?•<*• — 276 &d^P + 108 cdFi^f + 4800 efiP + 24 ed^«P + 420 ^d^P — 768 ed"^ — 3510 td^P + 360 edfgf — 1120 &^f — 700 d^p — 1516 «?«♦/ — 320 &d'f + 1112 d^eP + 900 (P^f + 2156 edV + 38 cW/» — 144 f^d^f + 8160 d^ — 430 dfiP — 108 i?*^f + 1620 C*(fo» — 2148 W^ + 336 #«•/ + 96 (MV — 1620 &6^P + 912 d'd^P — 40 #«• — 12 ed'ef — 864 ^d^^f — 15060 f^d^p + 2640 For the Nomerioftl Yerifioations see p. 806. 296 TABLES OP THE COVAMANTS M TO W OF THE BINABT QUINTIC. [143 V. No. 95 (continued). X coefficient. a' i'c'e*/ + 1840 a" Vd^p + • 184 a» 6*c»«/« - 594 «•<?«/» — 1280 d^P ■ 108 ed^p - 10296 <?^i?f — 13360 «v • • • (?d^f + 10080 c»<W + 3200 v<?p + 18 (?^ + 900 <?dfP + 7312 i?deP + 264 &d?^ + 19440 ^d^ff — 2360 <?^P + 756 C«(?«» - 8800 <?dP^ + 3840 c^P — 368 f^iPf - 9160 cd^^ — 5344 cd^^P — 732 i^d^ 11900 ed*(? + 2800 od^f + 540 c*d»« + 13900 rfy + 1956 ce« • • • c»rf» - 3150 iPt? — 1680 d^p — 1172 h^&dtp + 3564 6Vd/' — 36 df^f + 2520 c»«y - 1350 «•«•/» — 1296 rf*** — 1350 c»dV - 9540 <?dPeP + 1668 6'c*e/» — 144 i^d^ - 750 <*dtf/ — 1312 ^d?p + 376 ed^f + 4260 «•«• — 2060 <?d4?p — 1440 dd^i? + 10800 cV/* — 8020 (f^/ — 1530 c«d»c - 9100 cy«»/ + 15220 cW«/« + 6360 <*d? + 2000 c«dV + 1180 <»d^<?/ — 6000 b^c'P - 486 <»eP^ + 3712 i?d^ + 1350 ^'def + 1620 ifd'tf -- 8540 cd?P + 2344 c^V + 450 ^d^f — 2952 ed*^/ — 9260 &d^f - 720 <!•<?«» • • • edfif + 7200 &d^ - 2250 cd*a + 3330 <??/■ + 1720 d'd'e + 1800 <P — 810 <?«• — 1900 dd" - 400 wp • • • h\*d{p — 168 e'd'p — 576 «•«•/• + 648 e^f + 1824 t^tNp ~ 6420 i?d?p + 3792 cW/" + 9360 c»d'«y — 5808 cV + 450 tfd^ + 3240 «•#/» — 10100 <fd^/ — 4768 &dfff + 19920 <*d?^ — 6240 <?dW — 10300 (^dPf + 2608 cWe/ + 4920 c*^<? + 12440 cW — 10100 <?(Pe — 8160 cdff — 3440 <^cP + 1620 cdV + 7100 b'<»eP + 162 d»« — 750 (i'd'P ~ 702 wp + 36 c»rf«»/ — 90 ifdip + 2988 c««« — 1290 &i?f — 2880 c'dV + 1920 <fdfp + 14688 c'dV + 3640 <fd^i?/ — 22740 <fd'/ — 796 <*d^ + 600 &d^ — 5340 <*d^ef — 16520 fl»d»« + 3100 <!«(?«• + 23300 <j^d» — 600 edff + 8760 a* Vip + 18 «•<?«' — 5200 Vcdf< — 36 <?d!t — 5400 m^P — 180 <5rf» + 1500 143j TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. 297 V. No. 95 (continued). y coefficient. «• fdf* a» V'<?deP „«• 116 a« 6 V/ • • • a^ 6Vrf*ey + 7312 _. **/' e'^P + 80 6Vc/* — 20 i^d^ + 2344 «• *'«/• cd'P + 240 <^<Pf' — 280 i^d^P — 124 dtf* eePe'P — 160 <?d^p + 80 c*^ey — 8020 ^r ed^P — 120 i?e^p + 300 c*d^ — 10100 *Ve/» «•/ + 76 cd?ep + 160 C»(f»«/ + 3792 «P/» — 2 tfep — 120 edJ'^P • • • c>rf»«» + 14648 «fcy* + 4 d^e^P — 80 cd^f — 192 <?d^f — 702 a^P — 2 (P^f + 368 c^ — 108 &d?^ — 10296 dfef* + 6 dg — 180 <PP — 56 cd^e + 3564 df<?P — 16 V<*ep + 24 d}^P + 940 J" — 486 d^P + 14 <^cPp — 160 d^e'f — 1580 a^ 6V' + 6 * ^f — 4 <*d^P + 320 d"^ + 756 deP — 78 <»* iy« • • • <?eP — 280 6Vrf/* + 360 ^P + 72 *•«/» • • • ^d^eP — 560 cV/» — 420 h^(?eP — 44 d»/' + 2 ff^^P + 1280 &d^eP + 1440 cdPP + 332 d^P — 4 i?d<ff — 688 (?d^P — 2160 cd^P — 496 «y + 2 i?g + 184 <^^f + 984 c^P + 216 b'c'dp + 10 ctPp + 288 c^d^p — 480 d^eP + 304 e^P — 10 cd*^P — 240 &d^eP — 1320 d'i^P — 312 e<PeP — 26 ccPey — 480 (?d^e^f + 2040 d^f • • • ed^P + 32 «?«• + 264 (?d^ — 732 ^ • • • c^P — 6 dPtP — 144 cd^ep — 768 b*(^dp — 320 d'P — 30 d^<?f + 336 cd*^f + 2640 <?^P + 860 d*^P + 84 d*^ — 144 cd^^ — 1440 i^d^eP — 960 df(^P — 50 b'i^dp + 24 d^P + 504 &d^P + 1740 *•/ — 22 <f^P — 72 d^^f — 1296 <?^f — 2160 «» + 18 d'cPeP + 280 d^e'' + 648 cd'P + 420 wp — 6 (fd^P — 440 b'(^P — 108 cd^i^P — 2640 &d»P + 32 C*«»/ + 400 d'dep — 1296 cd^^f + 2910 e^P — 8 &d*P — 140 i?^P + 2344 cd(^ + 540 <?<pp + 4 ^df^P • • • d'd^P + 420 d^eP — 700 <M}i?P — 104 c»d«ey + 40 i^d'^P + 600 d^f + 1840 <?dt^P + 90 c»<fe« — 368 <^d/f — 3420 d^ — 1530 c"**/ — 26 c"d»«/» + 108 d"^ — 1172 h^^P + 96 cd*eP + 96 <?d'^f — 40 &d^ep + 900 d'dep + 80 edf^P — 160 c»d»«» + 376 i^d^f — 1280 f^^P — 3240 «?«•/ + 124 '<ap • • • ^d!"^ + 6360 &dp — 1120 C(*^ — 36 c^^f — 36 i?d^P — 576 &d^P + 3360 dFp — 36 cd?<^ — 168 i^de^f + 1668 ifde^f + 4800 df^P + 72 dPef • • • c«rfV — 6420 c>«« + 2520 d>t^f — 60 d?t? + 36 cdef .— 576 d'dep + 8160 __ <?«• + 18 o« 6»<^» + 6 cd^e^ + 2988 ^d^f — 13360 **• h'ep • • • «y' — 6 d^f + 162 i?d^ — 6000 Vcdp — 16 6V/» — 10 d»(5" — 594 cdp — 2288 c^P + 16 cdrf* + 180 b^<FeP + 432 cdt^f — 1312 d?ep + 8 c^P — 160 d'd^P — 144 cd*^ + 9360 d^P • • • tpp — 130 (^d^P — 1656 def + 1824 «•/' — 8 d»ey> + 60 cV/ — 1516 d^ — 2880 6V/» + 12 dt*P + 60 &d^eP + 912 h^d'eP — 72 For the Nmnerical Yerifioatioxis see p. 808. C. II. 38 298 TABLES OF THE OOVARIANTS M TO W OF THE BINABT QUINTIC. [143 V. No. 95 (concluded). y coefficient. a^ b^d'cPf* + 1620 a« h^P ^^ 276 a« 6*c«<fo* + 7100 &der + 3408 d^t^P + 908 i*d^ef + 12440 &^f + 2156 d}^f — 810 c-d*** - 5200 c'cPep — 15060 d^ ... <N^f - 5340 c^d^^f — 2360 6»cy* — 12 c'tPe' - 11900 c*cfe» — 9260 <^d^' + 832 &d?t + 10800 i?d'P + 4336 &i?P ■♦• 1244 c«d» - 2250 i?d^^f + 15220 i^d^P + 1112 h'd'ep + 486 (*<Pt^ + 19920 c'd't^P -• 168 i?d}p + 810 d'd^ef — 5808 <?de'f — 3510 i^d^f + 3330 <?iJ^f? — 22740 c«e« — 1350 c»«* - 750 cd^f — 90 cd^fp — 2148 ed^tf - 8160 Cd^i? + 10080 cd^^f + 3200 ed^i? - 5400 dl'e — 1350 cd'tfi + 1350 d^d^f + 3100 h^edp — 864 d^P + 1172 i^d^t? + 13900 e^P — 1008 d»«»/ — 2060 <*d^e - 9100 <^d?ep + 6624 c^«^ + 450 <jW + 1800 i^d^f — 5344 h'&eP — 240 h^i^^dp - 162 i^i? + 1720 <^d}p — 1620 e^^i^f - 810 &d>P — 1912 c'd^P — 3480 (^d}ef + 1620 &^i?f + 3712 <?*</• — 430 &df? + 1500 c»cr»«^ + 4920 <*d^ep + 2800 i^d^f - 600 i^d^tf — 4768 i?d^i^f + 3840 d^d^t? - 3150 d'd^ — 16520 &d^ + 7200 dd^e + 2000 &d^f + 1920 <?d^P — 2540 i^d? - 400 C«(?«» + 19440 i^d^i^f + 1180 c»d"« — 9540 c»cP«* — 10300 cdy" + 1620 Cd^if + 3240 6V/» + 162 cd»«» + 600 ^dep — 918 d^f — 1290 i»^f + 1956 d?i? + 900 ed^p + 240 V't^dp + 876 (?d^^f — 2952 c-eV + 1224 edi^ — 3440 <^d^^P + 2052 d^d^ef + 2608 &di?f + 2800 i*d^^ + 8760 &^ — 1900 &d^f — 796 ^d^P + 2100 &d^i? — 9160 d^d^i^f — 8540 c^d'e + 4260 c*cf«* — 10100 c»rf» — 720 (?d^ef — 6240 a' bp — 2 i^d'i? + 23300 l/cef* + 34 i?d!f + 3640 d"/* — 54 e^i? — 8800 d^P + 252 c<Pe — 750 eP — 216 d}' + 450 h^c'dp + 38 WP — 162 i?^P — 404 t?deP — 2304 cd'eP — 376 t?^f — 1680 cd^p — 216 i^d^P — 1560 c^f + 1080 &d^^f « • • 143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 299 W, 29 a. a»6«/» • « • o^ h^c^P - 1 5 a» 6Vdey* - 90 a» 6V(£«fiy + 1320 a»6'e/« • • • d^^P + 1< cV/» + 30 <?d^ - 260 b'cdp « « • (t^P - 3. 5 cd'ep - 210 f?d?p + 60 c^r • • • iP^P + 4( ctPe'P + 120 d'd^^P - 500 cPeP • • • d^^P - 1< cd'^p + 360 d'd'ep + 2235 <&»/• • • • dA?f - li cd^P - 420 (^d*d'f - 1995 «•/• • • • ^ + 5 cc»/ + 130 i?d^^ + 370 a' If dp • • • h^i^P — 1 d^P - 5 cd^ep + 360 ^r • • • c^deP + 1. 5 rfV/* + 195 cd?^P - 1320 1fi»p • • « c'i^P - K a d'eY* - 315 Cf^^f + 1110 edeP • • • &d?P • • d^d'P + 40 cd^^ - 210 e^P • • • c'cPe'P - 9< rf»ey + 165 d^P - 81 tPp • • • (^dk^P + 12< d^^ - 75 d^^P + 270 «?«•/• • • • i?^P - 4< 6V€/« - 10 d^^f - 225 dt/'P • • • i?d'tP + 61 c*rf»/« - 60 cfe« + 45 «•/' • • « f^d^t^p + 31 D i^d^p + 210 a* U'eP veep • ■ • f?d^^P - 18 c*«y* - 110 h^cdp tftpp • • • (?dep + 12 i?d^ep « • • ceP <?di?P • • ■ <?i?f - 2 f^d'^p 4 60 d^eP <*e*P • ■ • cd^P - 1 5 i?d&p - 360 d^P ed^eP • • • c^^p' - 11 <?^p + 240 ^r cd'^P • • • cd^eP + 26 5 c«rfy» + 30 h'i^p - 10 cd^p • • • cd'^P - 20 i?d^^p - 210 i?deP + 90 cey • • • cd^/f + 6 5 i^^ep - 180 d'^P - 60 d^P + 1 cde'^ - 1 &d}^p + 1140 cd^P - 120 d^^P - 5 d?eP + 4 5 i^d^f - 870 cd^i?P + 90 dPe*p + 10 d^e'P - 10 i?^^ + 130 cdt^P • • « ePe*/' - 10 d^i^P + 8 1 cd^ep + 310 c^P • • • <fc»/» + 5 d^e'f - 3 cd^d'P - 240 d'ep + 110 «"/ - 1 rf»«» + 5 cd^i^p - 390 d^^P - 50 cfVcP • • • a» h^p ccPey + 280 d^^P - 240 dtp • • « h'ceP ccPe* + 30 d^P + 280 ey • ■ • d?p d^P - 180 ^f - 90 V<?eP • • • d^P (fey» + 300 h^i^eP + 35 cd'P • • • eP dJ'^p - 120 i?d^P - 30 ed^P • • • h^i^dp d^^f + 30 (^de'P - 120 ofP • • • i?^P rf*e» 30 <?tP + 50 <?«/• « • « cd'eP \P(^dp + 15 <?d^eP - 60 dPt»p • • • cd^P &^P + 5 i?d^i?P « • • d^P • • • cd'P &d^eP - 30 i^d^P + 360 ^r • • • dp + i &d^p - 270 i^dp - 210 \f<*dp • • • d^^P - 4 &^p + 196 cd^P + 270 c'e'/' ■ • • d^^P + 6 c^d^p « • • cd^^P + 575 <»d*eP • • • d^P - 4 c^d^^p + 225 ccPtp - 1700 <?d^P • • • ^P + 1 (^d^tp + 615 cd^^P + 480 i?^P • • • h'd'P + 5 i^d^P - 660 cd^f + 670 cd*P - 15 c'deP - 6 c*ey + 45 ce" - 315 cd?<?P + 60 c't^P + 4 i^d'ep - 120 d^eP - 685 ccPey* - 90 i?^P + 9 &d^^p - 220 d^^P + 540 ««*?•/» + 60 c'd'e'p • i?d?^P - 980 d^i^P + 1515 For the Numerioal Verifications see p. 809. 38—2 300 TABLES OF THE COVARIANTS M TO W OF THE BIXART QUINTIC. [143 W, 29 A (contmued). a* 6»<i'«y 2080 a* 6^C(^^ ^^^ 615 a»6»rf»/» __ 196 a^6*(?e« - 3880 9f + 705 d}^ep — 945 dV/« — 660 h^dep - 300 b^(*df* + 110 d^f + 900 dP^r + 1840 i^d^p + 500 <^^r — 195 d"«» + 45 <?«•/» — 1040 d'd^P + 3810 d'iPef^ + 210 h^d^ep + 180 d^f — 180 i?^p - 3710 i^d^P — 575 edp — 60 «»»» + 216 ed^tp - 14040 i^i^P + 660 ed^p — 1420 h*<^df* — 265 &d^^P + 16120 c'd'f^ — 225 cV/» + 25 <?♦«•/• + 315 &d^P - 540 i^J^^P + 1350 i^d^ep + 780 <*(Pef* + 180 c»ey + 600 ^d^^P • • • i^d^^p + 5760 <»d^P + 1700 d'd^P + 7020 i^d^P — 1440 i*d^P — 2945 (!»«•/• — 1840 i^d't^P • • • c»ey — 75 (^ey + 1390 <?d>P — 615 d'd^e^P - 1950 (?d^ep — 1965 &d^P ... t?<Pi?/^ — 1350 i^d^^f - 17670 <?d^^P + 6000 &d^^P — 7020 ^d?/f* • • • C*(fo» + 4170 <?d^^P ^ 7050 &d^tP — 180 c'tfc'/* + 1560 &d^tp + 480 i?d^^f + 3000 &d}iff — 1275 <?*/ + 135 &^^P - 31040 i?d^ + 265 &d^ — 1110 cd^ef* + 2210 ed^^f + 45180 cd^P + 1420 c^d^eP + 3120 cd}<^P — 4100 <*d^d - 3160 cd'^P — . 3810 d'd^d'P + 3900 ccPfp + 6000 f?d^P - 140 cd^i^P + 2310 (^cy + 1240 ccP^f — 4880 i^d'^P + 18000 cd^^f + 1795 cV«' + 3155 oM + 990 i^d^^f - 12180 cd^^ _^ 1800 i?d^P — 515 <r/« — 25 <?d^i^ - 13430 ^eP + 240 i*d^P — 2920 d-^/* + 3710 cd^eP - 7200 d^d'P + 30 &d^e^f — 940 d««y* — 10755 cd^^f - 120 d^^f __ 870 c«d»e« — 4300 d^f + 9875 cd?^ + 9960 d^^ + 615 ^d^ep + 675 <?«• — 2845 d'P + 1890 h^eP _ 45 ed^^f + 510 6V/« + 100 rf^v/ - 540 i^dtp _ 310 i?d'^ + 2940 &def* + 240 d»e* - 1710 i?^P + 685 cd>P ... <!•«•/* — 540 b'ddp - 360 &d^P + 120 cd'^'e'f — 135 <*d}f* + 220 de'P - 240 i^d'e^P + 1965 cd^e' — 990 <^d>^f* — 6000 dd^P + 5840 &deP — 2210 rfiy • • • e*d6*P + 4100 dd^P - 6560 &^P — 960 c^V + 135 c***/* + 1340 ddp + 8460 e'd'ep • • « c^h'dp • • • ifd^P 4- 11700 dd^P - 3120 c'd^d^P -— 11700 ^P • • • t^dP^P • • • d^dp - 480 i^d^^P + 15435 h^i?P + 10 <*d^^P — 15240 dd^dP - 25880 i^d^f — 2760 CileP — 60 edgf + 6960 dddf - 1820 d"^ + 555 c^P + 40 &i? — 1620 dd - 3620 <?d^P — 780 d^P + 40 <»d*/* — 5760 dd^eP + 49680 &d^^P + 14040 d^^P — 30 c»d'«y» — 16120 dd'dp « « • i?d^tp — 10625 de'p • • • c'd'e'p + 26700 dd^df + 17520 i^dl'^f — 3220 ^P • • « <?d'<?/ — 5240 dd'd + 13500 &d}^ ^ 570 h^&eP — 40 c'rf"*' — 1640 dd^P - 120 (^deP — 5840 i^d'P + 180 ed?ef* + 6560 dd^dp - 32280 i?d^^P — 540 f^d^P — 360 od*^r + 7240 dd^df - 46880 &d^^f + 5550 c»«y* + 240 cd^ff — 24240 dd^d - 30040 c*rfV + 1285 cd^eP + 360 ed^d + 11420 dd^eP + 12860 cd^P + 990 cd^^P — 360 d^f* — 980 dd^df + 32000 cd^^p + 3150 cd^P • • • (P^P — 3420 dd'd + 46160 cdH^f — 3600 c^P • • • d?^f + 8100 dd'^P - 2700 143] TABLES OP THE COVARIANTS M TO W OF THE BINARY QUINTIC. 301 W, 29 A (continued). «» b^<NPi?f _ 8820 a» l^cd^^p + 1440 a» h^cd^d ^^ 18750 a« h^dd?d + 243000 C>(i»«* — 34620 cd^d'P — 1560 d^ep + 14115 dd'^ef + 2340 cd^tf + 1080 ed^P • • • d^df — 23790 dd^d — 89550 cd^t? + 12060 c^f • • • d?d + 8175 cd^f + 270 d^f • • • d^tp + 960 Wdp + 1320 cd>^d + 15120 d^e" — 1620 d^^P — 1340 ddp — 30 d^e — 810 b^c'T + 81 d^f^P — 2440 dd^ep + 540 b'd'ep + 945 edep — 990 d^df + 4320 dddp — 7240 dd^p — 675 f?^P + 980 dt^ — 1620 ddp — 20390 dddp + 7200 ^d^P + 515 \^&P — 81 dd^P — 3900 ddp — 14115 f?d?^P + UO d'dep + 390 dd^dp + 31040 dd^eP — 12860 ^d^p — 195 d'dp — 1515 dd^dp + 32370 dd^dp + 8220 c»ey — 5575 dd^P + 980 dddf + 38820 dddf + 150 ^d^tP + 120 dd^e^P + 7050 dd + 9310 dd + 6155 ed^^p — 800 ddtP — 6000 dd^ep — 49680 dd^P — 480 dd^i^f + 22600 d^P + 2440 dd*dp • • • dd'dp + 26700 ddd + 7240 d^ep — 15435 dd^df — 91260 dd^df + 63960 f^d^p • • ■ ddPdP + 15240 dd^d — 50550 dd^d — 6660 f^d^^p — 1260 dd^dp • • • dd^P + 800 dd^ep • • • d^d^/f — 42330 dddf — 6480 dd^dp + 81840 dd^df — 180600 c«^e« — 34340 dd + 1215 dd^df + 360 dd^d — 71610 f?d'tp + 480 ed^P + 2945 dd*d + 101450 dd^P — 4755 ffd^^f + 48360 cd^dp + 540 dd^ep — 8220 dd'df + 141240 C»(i»C» + 73828 ed^dp — 795 dd^df — 58080 dd^d + 219730 c*(iy» + 105 cd^df — 4180 dd^d — 34300 dd^ef — 45130 c^d^f — 30265 cd}d + 4185 cd}p — 7590 dd^d — 240975 C*(f<?* — 92290 d^ep — 8460 cd^df + 41640 dd^f + 5580 ed>^ef + 9540 d^dp + 20390 cd^d — 4650 dd'^d + 128490 c»(i»e» + 69220 dPdf — 16194 d}^ef — 5580 dd^e — 34155 c»(P/ — 1215 d^d + 3765 d}^d + 1980 cd"* + 3645 c»rf"«« — 30510 h^dep + 120 b'dp — 270 h^d^dp • • • crf»« + 7290 dd^P — 2235 ddep — 3150 d'dp — 1890 — 729 dddp — 2310 ddp + 3420 d'd*ep + 2700 ^^\?cP — 5 ddp + 10755 dd^P + 2920 d^ddp + 7590 deP + 15 dd'eP + 10625 dd^dp — 18000 d^df + 8256 /^' — 10 dd^dp — 26700 dddp + 43800 dd'P — 105 IPc^P + 10 dddp + 795 ddf + 5030 dd'dp — 14360 ed'P — 120 ddf — 10070 dd^eP + 32280 dd'df — 43605 cd^P + 420 dd^P + 180 dd^dp — 81840 ddd — 12310 c^p — 280 dd^dp + 1950 dd^df — 85800 dd^ep + 4755 d^ep — 240 ddPdp • • • ddd — 28710 dd'df + 77790 d^^p + 210 dd^df + 36510 dd^P + 1260 dd^d + 59835 df?P • • • ddd • • • dd^df^ « • • diC^f^ • > • Jf" • • • dd^ep + 25880 dd'df + 181980 dd^df — 57060 \^edp + 200 dd^dp — 32370 dd^d + 153480 dd^d — 114960 &^p — 40 dd^df — 12180 dd^ep — 26700 dd^ef + 19020 ^d^ep — 1140 ddFd — 9850 dd^df — 41360 dd^d + 109660 ^deP — 480 cd^P + 195 dd^d — 306900 dd}^f — 2481 e^p + 1040 ed'dp — 43800 dd^P + 14360 dd^d — 56110 cd^P + 660 cd^df + 72755 dd^df — 16170 dd}^t + 14895 302 TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. [143 W, 29 A (continued). a» 6 VeP _^_ 1620 a> b^d^P + 5575 a^ h'cd^^ + 41250 a> b^<^P + 5580 I a' b'^f + 1 d'e'P — 5030 d^f + 5445 c'^d^P — 9540 \ b^ce/^ + 10 d^t'f — 4255 d}^^ . — 6525 c'^d'ey* 2340 1 d^P + 20 d^^ + 2175 }^<?ep — 900 i^'^d^f + 20610 1 d^P — 130 b^i*dp — 1110 <?d}p — 510 c^V — 4350 1 ^p + 90 cV/* + 870 (»dep + 120 <?d^P + 45130 1 b^i^ap — 65 &d^ep — 5550 ^t'P + 23790 d'd^d'f — 92200 c»ey» — 165 <*df?P + 24240 ed^tp — 32000 &d^d^ — 25050 cd^eP + 870 &^P + 16194 f?d^^p + 58080 e^P — 19020 cd^P — 670 d'd'P t- 1240 ^d^f — 15440 i^d^^f + 46050 Ci^P + 180 i^d^^P — 45180 cV — 12500 cWj* + 138750 d^P — 45 d'd't^P + 12180 <^d^P — 48360 dd!ef • • • d^e^P + 75 i^d^f — 66650 i^d'i^P + 41360 dd^i? — 178200 d^e^P — 135 cV — 8550 d'd^^f — 181600 ^d?f — 1650 d^P • • • &d^eP — 17520 c»^^ — 18400 c»cf^«» + 103950 ey • • • <?d'^p + 91260 &d^eP + 180600 &d}^t — 30250 b'c'P + 30 c'd^e^f • • • &d^i?f • « • c*cP + 3600 <?deP — 280 c'd'e' + 62100 c»c^e» + 289800 b^i^eP • • • <?i?P + 2080 i^d?p — 22600 c'd^P — 77790 d^d'P + 1215 &d?P — 1320 i^d^^P + 85800 i^i?f — 87000 d^d^P — 270 i?d^^P — 3000 i^d^e'f — 148890 c*rf««* — 318500 d^f — 5445 i^de^P + 4880 c'd*^ + 1850 <?d^ef + 92200 c^'cPeP — 5580 i^i^P — 4320 cd^ep — 150 c'd'^ + 179500 c"cW/ + 17520 cd^eP + 2760 cd^d'f + 15440 c'd''/ — 17520 c"cfo» + 8700 cd^^P — 6960 cd^^ + 10350 c'd'^^ — 69000 d'd^p + 2481 cd^^P + 6480 ^ioy2 — 8256 cd^e + 15300 d^d^i?f — 10595 cd^f • • « <^^/ + 12210 d"* — 1350 <^d^^ — 31150 c^ • • « c/»c^ — 7050 b^c'^dp + 135 dd^tf + 1650 d^P — 1390 b'^P + 225 c^^^P + 540 i^d^d + 37950 d^^P — 600 c'deP + 3600 &d^ep + 8820 dd^f • • • d*^P + 10070 c'^P — 8100 (?d^P — 41640 ddPt? — 22275 d^i^f — 12600 (^d^P + 940 i^f^f — 12210 ddf'e + 6600 c^e» + 4050 i^d^^P + 12180 i»d*P + 30265 dd^ — 800 b^^ep — 30 d'ilep — 72755 &d?^P + 16170 a« &"«/• — 5 i^d^P + 1995 c«.y + 4255 i^d^e'f + 62025 b'^edp + 10 d'd^P — 1795 (*d*ep + . 46880 d'dt^ + 44225 edp + 75 c^eV — 9875 <^(P^P — 360 dd^ep — 141240 d^fP — 130 i c»c/»c/* + 3220 (^d^/f + 148890 dd'^f + 87000 ddp + 315 i?d?^P + 5240 &d^ + 38950 dd^^ — 129000 dp — 216 &d^P + 4180 c^'d^P + 42330 i^d'P + 57060 l^dP . — 5 \ <^eV + 12600 i^d^^P • — 181980 d'd^^f • • • ddep — 30 ! c'dJ'P + 1275 c'dVf • • « c»c^«* — 5250 ddp — 705 i?d^^P + 17670 c*d'd' — 220125 dd^ef — 46050 cd^P + 260 c'd^ep — 36510 &d^eP — 63960 &d^i? + 122800 ed^dp — . 265 &d}^f • • • &d^^f + 181600 e'dy'f + 10595 ed^P — 990 c^d^ — 6075 ed^^ + 159000 c^d^^ — 88125 c^P + 1620 cd^eP + 1820 f?d^P + 43605 i^d'^e + 27300 d^P ._ 555 cd^^p — 38820 i?d^^f — 62025 c«cP — 3375 d^P + 1620 cd^^f + 66650 c'd'e' — 92500 b^^P • • * d'dP — 1215 cd^e' — 19800 Cd^'^ef — 20610 c^^deP — 1080 ddf « • • 143] TABLES OF THE COVAEIANTS M TO W OF THE BINARY QUINTIC. 303 W, 29 A (concluded). »<^ w • • • a« 6V(^/» + 34340 d^ h^i^d^ef — 138750 a<» 6 We» - 17875 Wtp + 30 &d^i?P — 153480 c^dd' — 1250 ddf - 6600 ed^p — 370 c»c^ey + 220125 i?d}^f + 31150 dd'd + 4125 edi^f' + 1800 i?di^ • « • d'dJ'd' + 40000 dd^e • • • ^t'p + 2845 i?d?eP + 6660 c'd}^e — 18750 dd^ • • • c»d»e/* + 570 ^d^f + 18400 cd^ + 2250 h^d^P + 729 c»(^ey» + 1640 i?d^^ — 73375 U'c^^P — 135 d*dep - 3645 ^d^P — 4185 cd^P + 12310 c'^'dep — 12060 d^df + 1350 i?^f — 4050 cd^e^f — 44225 c^V/» — 1980 d'^dp + 1620 cd^P + 1110 cd'f^ + 42500 &d^P — 69220 d^ddf + 3375 cd^^P — 4170 d^'ef + 4350 d'd'^P + 89550 d^dd - 2250 cd*t^P • • • d»e» — 5125 d'de'f — 41250 d^def - 3600 cd^^f + 6075 WeP — 45 d'd' + 5125 d^dd + 2125 afo" • • • ed^p — 2940 ed^eP + 240975 d^df + 800 d^eP + 3620 ed^p — 9960 d'd^^f — 179500 dH^d - 500 ^^P — 9310 <?eP — 8175 d^d^d' + 80125 d^de • • • rf*cy + 8550 (^dJ'eP — 46160 dd^P — 109660 dd • ■ • ^^ — 3375 d^d^d'P + 34300 dd^f — 122800 Vf^dp + 210 i^d^f — 10350 c'rfV + 1250 ff^p — 615 i?^ + 7375 d'def + 178200 f^d^ep — 1285 &d^P — 73828 d'd^^ • • • c'd^P — 11420 &d>i?P + 306900 dd^f — 37950 c*cy» — 3765 &de'f — 159000 dd^d" — 37125 i^d'P — 3155 i^d^d' + 73375 d'd^^e + 17875 &i^i?P + 3160 i^d^ep + 71610 d'd^ — 2125 &d^^p + 9850 &d^^f — 289800 ¥d^ep + 1620 (^d^f + 19800 <^d^^ • % % d'dp + 30510 &^ + 3375 ^d^P — 59835 d^dd'P — 15120 ed^tp — 13500 (^dd^f + 129000 d'i^f + 6525 ed>^p + 50550 C>(^6* + 80500 d'^d^ep — 128490 ed^^f — 62100 ^d^tf + 25050 d'^d'e'f + 69000 ed"^ • • « c«^e» — 80125 d^dd — 19875 cd^P — 7240 cd}'f — - 8700 ddp + 56110 c^^P + 28710 crf^V + 19875 dd^ey + 88125 cd^t^f — 38950 cf'e — 1125 ddd — 40000 cd^^ + 25875 h^dp + 990 dd^ef — 103950 d^eP — 6155 i^^P + 1710 ddd + 37125 (fey + 12500 ^d'ep + 34620 dd^f + 22275 d*^ — 7375 f»d^P + 4650 ddd « • • h^ep — 45 i?^f + 7050 dd?t — 4125 d^dep + 615 ed'P + 92290 ddP' + 500 c»«y» + 3880 ed^^p — 243000 h'd^dP — 7290 &d^P + 4300 ed^^f + 92500 d^dp + 810 &d^^P + 13430 edd^ — 42500 d^d'eP + 34155 &d^P + 18750 <^(^ep — 219730 d^ddf — 15300 &^f — 2175 i*d^^f + 318500 d^d + 1125 i^d^eP + 30040 d'd^ — 80500 d'd'P — 14895 ^d^e'P — 101450 &d:^p + 114960 d'd^df — 27300 t^d^^f — 1850 &^^f + 5250 d^d^d + 18750 d'dfP — 25875 c»cfe* « • • d'^dtf + 30250 304 TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. [l43 For the lower covariants the numerical verifications are given for the entire coeflScient, but for the higher ones where the number of terms in a coeflScient is con- siderable they are given separately for the diflferent powers of a; and it is also interesting to consider them for the separate combinations of a and b. I recall that the positive and negative numerical coefficients are summed separately, so that (+a number) means that the sum of the positive numerical coefficients is equal to the sum of the negative numerical coefficients and thus that the whole sum is =0. It is to be observed that for the lower covariants the sums of the numerical coefficients do not vanish for the separate powers of a: thus in the invariant 0, 141, the sums of the numerical coefficients for the terms in a\ a^ a® are =1, — 2, 1 respectively. As regards the invariants Q and Q\ for the first of these, Q, the sums of the numerical coefficients for the terms in a*, a', a', a\ aP are each of them =0, but this is not the case as regards Q' ; in fact Q' is = G' 4- a multiple of Q ; hence the sums for Q are the same as those for G*, viz. they are =1, —4, +6, —4, +1 respectively. Like results present themselves in other cases, and they might probably be accounted for in a similar manner; we have a series of sums not each =0, but which are equal to a set of binomial coefficients taken with the signs + and — alternately and thus the sum of these sums is = 0. For R, 8 and 8\ I have given the sums for the different powers of a; and in regard to iS I give here the following paragraphs from the Tenth Memoir on Quantics : — I remark that I calculated the first two coefficients So, Si, and deduced the other two, Sa from 8i, and S, from 8o, by reversing the order of the letters (or which is the same thing, interchanging a and /, b and e, .c and d) and reversing also the signs of the numerical coefficients. This process for flfj, flf, is to a very great extent a veri- fication of the values of So, 8i. For, as presently mentioned, the terms of So form subdivisions such that in each subdivision the sum of the numerical coefficients is = : in passing by the reversal process to the value of flf,, the terms are distributed into an entirely new set of subdivisions, and then in each of these subdivisions the sum of the numerical coefficients is found to be = 0; and the like as regards Si and S^. If in the expressions for So, S^ 8^, S, we first write d = e=f=l, thus in eflfect combining the numerical coefficients for the terms which contain the same powers in a, 6, c, we find So = a' (- 2c» + 6c» - 6c 4- 2) -h a« {6»(6c« - 12c - 6) + 6(- 15c' 4- 33c»- 21c 4- 3) 4- b' (42c* - 147c» 4- 195c" - 117c 4- 27)} 4- a {b*. 4- 6» (30c* - 36c 4- 6) 4- 6' (- llTc* 4- 249c« - 183c 4- 51) 4- 6 (9c« 4- 148c* - 378c' 4- 330c» - 99c) 4- b' (- 63c« 4- 166c* - 147c* 4- 45c»)} 143] TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. 305 + a^{6^2 + 6»(-15c + 3) + 6*(76c>-69c + 24) + &»(-9c*-167c'-i-225c«-87c-2) + 6» (72c» + 48c* - 186c» + 96c») + 6 (- 126c« + 201c» - 87c*) 4- 6* (27c« - 4f6c' + 20c«)} which for c = 1 becomes = 26« - 126» + 306* - 406» + 306« -126 + 2, that is 2 (6 - 1)«, and for 5 = 1, becomes =0. 8, = a» (Oc» + Oc + 0) + a« {6» (Oc 4- 0) + 6 (3c^ - 9c» 4- 9c - 3) + 6* (24c* - 99c» + 163c» - 105c + 27)) + a {6*.0 + 6'(-6c« + 12c-6) + 6''(-24c« + 90c''-108c + 42) + 6 (33c* - 90c» 4- 54c» 4- 30c - 27) 4- 6* (- 27c« 4- 78c« - 66c* 4- 6c» 4- 9c»)} + a« {6* (3c - 3) 4- 6* (- 1 5c 4- 15) 4- &» (6c» - 1 2c'» 4- 36c - 30) 4- 6' (9c» - 42c* 4- 84c» - lOBc^ + 57c) 4- 6 (9c« - 54c» 4- 96c* - Sic*) 4- b' (9c' - 9c«)} %vhich for c = l becomes =0. /S% = a» (Oc 4- 0) 4- a» {6» . 4- 6 (Oc« 4- Oc 4- 0) 4- 6' (18c* - 72c^ 4- 108c* - 72c 4- 18)} 4- a l&»(0c4-0)4-6»(-33c» + 99c«-99c4-33)+6(57c*-162c»4-144c«-30c-9) 4- 6" (- 60c« 4- 207c* - 261c» 4- 141c" - 27c)} 4- a« {6» . 4- 6* (15c« - 30c 4- 15) -f 6» (- 54c» 4- 102c« - 42c - 6) 4- 5» (123c* - 297c» -f 243c« - 87c 4- 18) 4- 6 (- 27c« + 102c* - 96c» 4- 21c=) 4- 6" (27c' - 60c« 4- 51c« - 12c*)} '^'V'hich for c = l becomes =0. ^= a».0 4- a* {5 (Oc 4- 0) 4- 6« (Oc» 4- 0c» 4- Oc 4- 0)} 4- a {6».0 4-5»(0c"4-0c4-0)4-6(-9c*4-36c'-54c*4-36c-9) 4- &• (36c» - 171c* + 324c» - 306c« 4- 144c - 27)} 4- a' {6* (Oc 4- 0) 4- &» (7c» - 21c« 4- 21c - 7) 4- 6»(- 39c* 4- 135c'- 171c» + 93c - 18) 4- b (66c* - 243c* 4- 333c» - 201c» 4- 45c) 4- 6^ (- 27c' 4- 101c« - 141c« 4- 87c* - 20c»)} '^^liich for c = l becomes =0. It follows that for c = rf = e=/=l, the value of the co variant S is =2(6 — 1)V, "^^liich might be easily verified C. n. 39 306 TABLES OF THE COVARIANTS M TO W OF THE BINABT QUINTIC. [143 For T, U, V and W, I look at the sums for the different combinations of a and £ Thus for T we have a; coefficient. y coefficient. a*b^ 26 26 a^b'^ -= 12 a»6> * 14 M\/ a»6« * 2 b^ 141 6* 112 b^ 281 436 6« 281 aH* ^ 1 zUV/ a»6» A 42 6» 106 6» 546 6« 186 b' 696 b' 1173 b' 366 b' 2272 3738 a^l^ * 16 *j t tf\j a} I/' * 5 6< 359 b* 179 6» 1411 b* 821 6« 3103 6' 2097 b^ 3030 b" 2147 b^ 1197 9116 b' 1262 a«6' «^ 2 «7 X X V 6« 92 — 78 an« A 28 6» 307 — 349 6» 342 6^ 1073 — 1003 b* 1790 6» 2040 — 2110 6» 3496 6« 1930 — 1880 6« 3445 6> 1207 — 1221 6> 2064 6« 231 — 239 6880 6<> 463 -i 12 395 1650 6511 11628 A 20196 ^ 20196 Observe here that in the ^^-coefficient for the terms in a® the successive sums are -2, 4- 14, -42, +70, -70, +42, -14 + 2, which are the coefficients of -2(tf-iy. TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 307 ^r U we have ^ 36 ^ 24 b' 198 b' 242 ^ 2 6' 208 b' 286 b' 866 6^ 1246 ^ 64 b* 328 6» 1258 b^ 2586 b' 2186 b' 856 dc 4 b' 70 6» 448 b* 1488 6» 2140 6= 1678 b' 884 b' 166 36 464 2608 7278 6878 17264 39—2 308 TABLES OF THE COVABTANTS M TO W OF THE BINARY QUINTIC. [143 For V we have X coefficient. a»6« ^ 36 a*b* ^ 20 b' 284 b' 1094 a'b* 2 6» 184 6» 1656 b' 3624 b' 4898 an» * 14 6* 666 6» 6608 6« 10512 b^ 22042 b^ 9162 a^b"" 4 6« 76 48 6» 2956 - 3040 6* 11946 ^ 11806 6» 23924 - 24064 6« 25110 - 25026 b' 25524 - 25552 6« 8822 - 8812 a«6« 18 6^ 184 324 6« 4098 - 3622 6» 19350 - 20274 6* 42398 - 41278 6» 51872 - 52740 6« 44320 - 43900 6^ 20624 - 20740 b^ 3870 - 3856 36 1398 10364 49004 =t 98358 db 186734 345894 a»6« b"" y coefficient. ^ 24 4 144 436 a»6» ^ 24 6» 776 ¥ 2696 ¥ 1264 a»6» ^ 6 b* 300 6» 2236 6« 8616 ¥ 15442 ¥ 33044 a}b' ^ 78 6» 852 b* 8310 b^ 30200 6» 56740 b' 39956 b' 17986 aPb^ 2 V 286 — 270 6« 2026 — 2082 6» 9360 — 9248 ¥ 19760 — 19900 b^ 36442 — 36330 b* 30340 — 30396 b' 23426 — 23410 b' 5120 — 5122 24 584 4760 59644 154122 126760 345894 Here in the ^^-coefficient for a^ the successive sums are — 4, +28, — 84, 4- 140,- — 140, + 84, — 28, + 4, which are the coefficients of — 4 (tf — 1)' ; and for a® the successive sums are 18, -140, +476, -924, +1120, -868, +420, -116, +14, which are the coefficients of 18 (tf — 1)^ + 4 (tf — 1)'. In the y-coefficient the successive sums are -2, +16, -56, +112, -140, +112, -66, +16, -2, which are the coefficients of -2(tf-l)». ] TABLES OF THE C0VARIANT8 M TO W OF THE BINARY QUINTIC. 309 Finally for W we have 2972759 7A0 a' 6 16 aH -fc 175 h^ 806 a^h" * 80 6« 1175 6» 2760 ¥ 6871 a*b* Ik 570 b^ 5200 6« 18005 6* 44720 b' 23810 a*6* 't^ 90 6» 2386 b* 26675 6* 84680 6« 107730 b" 199160 b' 240499 a«6' * 15 6^ 640 &• 8260 6» 59135 6< 182055 6« 341470 6» 699260 6» 612015 b^ 304501 16 981 10886 92305 661220 a^6»« + 1 6^ 120 - 130 6» 1125 - 1080 6' 30350 - 30470 6« 122400 - 122190 6» 332494 - 332746 6* 729150 - 728940 6» 880750 - 880870 6« 466935 - 466890 6^ 363670 - 363680 60 76116 - 76115 a«6" + - 5 b'^ 400 - 346 6» 3500 - 3765 6» 26240 - 25460 6^ 154030 - 155560 6« 409700 - 407600 6» 747985 - 750043 6* 745920 - 744480 6» 613100 - 613805 6» 311790 - 311560 6^ 89215 - 89260 bP 9999 - 9995 3003111 3111879 9087749 2207351 Here for the terms in a* the successive sums are 1^ _10, +46, -120, +210, -252, +210, -120, +45, -10, +1, ;h are the coefficients of (tf- ly^ and for the terms in a* the successive sums are «5, +54, -.265, +780, -1530, +2100, -2058, +1440, -705, +230, -45, +4, ± are the coefficients of - 5 (tf - 1)" - (« - ly. 310 [144 144. A THIRD MEMOIR UPON QUANTICS. [From the PhiloaophiccU Tramactiona of the Royal Society of London, voL cxlvi. for the year 1866, pp. 627—647. Received March 13,— Read April 10, 1856.] Mt object in the present memoir is chiefly to collect together and put upon record various results useful in the theories of the particular quantics to which they relate. The tables at the commencement relate to binary quantics, and are a direct sequel to the tables in my Second Memoir upon Quantics, voL CXLVI. (1856), [141]. The definitions and explanations in the next part of the present memoir are given here for the sake of convenience, the further development of the subjects to which they relate being reserved for another occasion. The remainder of the memoir consists of tables and explanations relating to ternary quadrics and cubica Covariant and other Tables, Nos. 27 to 50 (Nos. 1 to 50 binary quantics)^ Nos. 27 to 29 are a continuation of the tables relating to the quintic (a, 6, c, d, e, fjx, yf. No. 27 gives the values of the different determinants of the matrix ( a, 46, 6c, 4d, e ) a, 46, 6c, 4d, e 6, 4c, 6d, 4«, / 6, 4c, 6d, 4«, / determinants which are represented by 1234, 1235, &c., where the numbers refer to 1 The Tables 49 and 60 were inserted October 6, 1S56.— A. C. 144] A THIBD MEMOIR UPON QUANTICS. 311 the different columns of the matrix. No. 28 gives the values of certain linear functions of these determinants, viz. 2,= 1256 + 2345- - 2 . 1346, L'^i . 1256 - 1346, 8M = - • 1345 + 2 .1246, 8ir=- 2346 + 2 . , 1356, 8iV=- ' 1245 + s . 1236, 8N' = - 2356 + 8 . . 1456, 80P = L'-3L = 8. 1346 - 8 16P' = - 5L'- L = — 18. 1256 - 8 ^t the end of the two tables there are given certain relations which exist between tr^he terms of Tables 14, 16, 25, 26, 27 and 28. No. 27. 1234. 12d6. 1236. 1246. 1246. 1345. 1256. 2345. «V aV- ^ a*d/+ 6 a'd/- 6 a V + 4 a'c/" ay«+ 1 ay« «»« - 16 a^de + 24 aV aV + 16 oi^- 4 a6^- 24 o^- 2 a6«/* «V + 36 aiy+ 4 abc/^ 22 abr/+ 6 a6e«- 4 abe^ + 64 occf/"- 16 acd/+ 20 i ^3i«e+ 16 abee- 84 abde- 6 ahde- 26 o^y- 24 acy+ 24 octf« + 16 ace* - 80 ; ^3bcd- 152 oW"- 24 ac^e + 16 ac«« - 96 acde-\- 24 acde- 208 acPe + 16 ad^e+ 60 «j* + 96 a(j«rf+ 64 acd^ acd^+ 96 ocP ew^ + 144 6V/- 15 6*^- 80 *(i + 80 b^e + 60 by + 16 6y ... 6»c/ + 24 6V 6»62 ... 6V + 240 ; «V - 60 i^cd- 40 b'ce - 10 b^ce + 90 bHe - 20 6»c^- 40 6cy ... be"/ + 60 ; OCT ... 6«rf« ... 6»c?» - 80 oc^c ... 6c»6 + 60 6crfc 6cde- 860 1 bc^d bc^d 6ccP bed"- 40 6rf» 6c^ + 960 ■ 1 C/ • . . (j . • • c»c/ c'c/ c»« c»e + 960 j i 1 c'd' ... c'd' - 320 j 1346. 2346. 1356. 2356. 1456. 2456. 3456. ay* ... ai^ a^+ 4 acP- 6 axp + 6 a<^«- 4 a^" a6c/'+ 16 ac«/*- 24 cLcef — 4 adtf-v 6 adef- 22 acy+ 4 6(^- 16 acc(/'- 36 a(^/+ 24 a(/y- 24 ae* a<j» + 16 6c/« + 24 6«y + 16 ac«« - 16 a{2s* a^ + 24 ^»y« + 16 6y* ... bdef-^ 84 cy« + 36 ad^e + 36 6««/+ 64 V'tf- 4 6cc/- 26 bcef^ 6 ftc» + 60 cdef- 152 l^df-- 16 bcdf-^ 208 6<j(^+ 24 6«((/-- 96 6c^/+ 16 c»c/ - 24 ce» + 80 6V ... 6<j«« - 40 6ce« - 20 6de» + 90 6<^- 10 ccZy+ 64 c^/ + 96 6cy+ 36 6rf«e + 60 bd^e i^df + 96 d'df ... cc^ - 40 d?^ - 60 ! bcde- 20 c»/ + 144 cy ... e^ - 80 vTVi ... d^t 1 W» c»cfe- 40 c*cfe ccPtf cc^e V c ... CflP CflP cf* d^ <?d^ 1 312 A THIRD MEMOIR UPON QUANTICS. [144 No. 28. N. M. L. L'. P. F. M'. N'. aHf + 3 a V - 2 ahcf- 9 abde+ 1 a V + 1 abd/+ 2 a6c»- 9 ocy- 9 ay « + 1 a6c/- 34 aor//*+ 76 acc« - 32 ay«+ 3 oic/- 22 acdf- 12 occ' + 64 ay> ... ahef+ 1 oo//- 3 oc^ + 2 ay«- 1 aA«/'+ 9 occ^- 1 aci^ ^ 18 abf-^ 1 ac«/*+ 2 (w^/- 9 a<i«»+ 6 ac/^+ 3 ^ oic/- 9 oc* + 6 \ 6y«- 2 cu^e + 18 acde-i- 32 ad'e- 12 a<Pe- 36 cuPe o(f"e+ 12 bhf- 9 bee/ + 1 acd^- 12 b*/ + 6 o^ - 18 6 V + 6 6V/- 32 6V + 225 6V/+ 64 6V - 45 b'd/+ 2 6V - 9 b^df- 18 6V 6«i/'+ 32 6dy+ 181 W«» - 1^ ; b^ce - 15 b^d" + 10 b^de bc'e - 15 bc^f- 12 6cfl?c- 820 bey- 36 6ccfe+ 20 bc\f ... 6<;flfe+ 31 6cy + 12 ftcSc + 45 bd'e- 15 cy - 18 c^- 12^ \ C*(5« + lO^ ! 6c'cZ bcd^-^ 10 b<P + 480 M» bd* - 18 6ii» - 30 c'de + 10 cd^e ... * c* C^(/ c^e + 480 CJ C/ • • • c«e - 18 c'e - 30 ci* c^ ... -' c'd' - 320 c"flP c'd' + 12 c«cZ« + 20 If the coeflBcients of the table 14 are represented by ^A, B, ^C, viz. ¥rritiDg il = 2 (oc - 46d + 3c»), B = a/- 36e + 2cd, (7 = 2(6/-4ce+3d«), then we have the following relations between 1234, &c. and A, B, C, viz. 1 + Bx + Ax 1234 = + 6a« -12a6 + 16 oc -10 6» 1235 = + 6 ah - 2 oc - 10 6> + 6ad 1236 = - 2ac+ 8 6= + 6<k£- 18 6c - 2d/+ 8c» 1245 = + 18ac - 6 CM? - 30 6c + Sae +10 bd 1246 = + 12 6c + 4a<5- 4 6rf-24c« + 46c + Scd 1345 = + 24 ac£ - 8 oc - 40 6rf + 4 a/ + 20 6c 1256 = - I ae + 4 6rf + 3c» + I a/ + 5 be - IS cd - lb/ + 4cc+ 3dr 2345 = + 20 oc + 40 6c/ - 30 c« - SO be + 20 cd + 20 6/ +40 cc- 30 (? 1346 = + 4 06 + S bd + 6c» -36 erf + 4 6/+ 8cc+ 6d^ 2346 = + 4 a/ + 20 6c - 8 6/- 4cc + 24c/ 1356 = + 4 6c + 8 cc? + 4 6/- 4c«-24rf« + 12cfe ' 2356 = + 8 6/ + 10 cc - 6c/-30cfo + lSd/ 1456 = + 6cc + 6 c/ -ISde - 2c(^+ 8c» 2456 = + 6 c/ - 2((/*-10c» + 6 c/ 3456 = + 16 df- 10 e^ -12 c/ + 6/« and the following relations between L, L\ &c. and A, B, C, viz. Cx + Bx + Ax ^= - 3 oc + 3 6« + 3 orf - 3 6c - lac+ 1 6rf 3/ = - Sad+ 3 6c + 3ac - 3c' - l€/+ led Z = + 11 oc + 28 6rf-39c» + 1 a/ - 75 6c + 74 cd + 11 6/+ 28 cc- 39 rf« Z'- - 7ac+ 4 6rf+ 3c' + 3 a/ + 15 6c - 18 erf - 76/+ 4cc+ 3rf» 2iP = - lac- 2bd+ 3c» + 3 6c - 3 erf + 1 h/+ 2 ec- Srf" 7^ = + 3ac- 6bd+ 3c« - 1 a/+ 1 erf + 3 6/- 6 oc + 3 rf» J if'= - I a/ + I cd + 3 6/- 3rf« - Sc/+ 3rfc I jr= - 16/+ 1 cc + 3c/ - 3rfc - 3(i^+ 3c» 1 _ iiiJiMOIIl UPON QUANTICS. 313 -» e nave also the following relations between i, L\ &c. and a, 6, c, d, c, /, viz. aP -6Jf 4- ci^T =0, aN'+2bM''-cr + 3eN = 0, = 0, 3cJ\r'-2dif'4- eF+fM=0, The quartin variant No. 19 [G] is equal to i.e. it is in fact equal to —4 into the discriminant of the quintic No. 14, [A]. The octin variant No. 25 [Q] is expressible in terms of the coefficients of Nos. 14 and 16, viz. A, B, C, as before, and Ja, /8, 7, JS the coefficients of No. 16, [D], i.e. a = 3 (ace -ad^- h^e +26c(Z- c»), /8= acf-ade- l^f+ bd^ + bee - d^d, y = adf—ae^ — bcf+ bde + d^e —cd^, S = 3 ibdf" 6e» 4- 2cde - c*/ - d»), then No. 25 is equal to A, B, C a, ^, 7 The value of the discriminant No. 26, [Q'], is (No. 19)«-128 No. 25. [that is Q' = G« - 128Q.] We have also an expression for the discriminant in terms of Z, L\ &c., viz. three times the discriminant No. 26 is equal to [or say 3Q' =] LU 4- UMM' - &4^NN\ remarkable formula, the discovery of which is due to Mr Salmon. It may be noticed, that in the particular case in which the quintic has two square ^tors, if we write (a, 6, c, d, e, /$a?, y/ = 5 {(p, 5, rjx, y)«}« . (\, /A$a?, y). C. II. 40 314 A THIBD MEMOIR UPON QUANTIC8. [144 then a = 5\p^, h = 4tpqK 4- J5 V, and these values give P =Z(63»-pr), M = K, lOpq, where the value of iT is c = (23" +pr)\ + 2pqfi, d = 25r\ + {2f +pr)fi\ 8 (pfi^ - 2qfi\ 4- rVy (/)r - j«)*. The table No. 29 is the invariant of the twelfth degree of the quintic, given i^ its simplest form, Le. in a form not containing any power higher than the fourth o^ the leading coefficient a: this invariant was first calculate by M. Faa de Bruno. No. 29. [See U. No. 29, p. 294] The tables Nos. 30 to 35 relate to a sextic. No. 30 is the sextic itself ^ No. 31 the quadnn variant ; Nos. 32 and 33 the quadricovariauts (the latter of them, the Hessian); No. 34 is the quartinvariant or catalecticant ; and No. 35 is the? sextinvariant in its best form, i.e. a form not containing any power higher than the- second of the leading coefficient a. No. 30. a^\ 6 + 6 c + 15 d+20 e + 15 /+6 g+l Wx, y)* No. 31. No. 32. ag + 1 ¥ — 6 ce + 15 d^ — 10 ^16 a« + 1 W - 4 c« + 3 a/ + 2 6tf - 6 cd + 4 0^+1 C6 - 9 (^ + 8 hg -^ 2 cf - Q de + 4: eg + I ^/- 4 e» + 3 :fe4 ±G No. 33. *9 ^6 5«, 2/y ac + I ad+i otf + 6 a/+ 4 ag+ 1 6^+4 eg + ^ dg + 4: eg + 1 ("-' 6c -4 bd+ 4 6<»+16 6/+ 14 C/ + 16 d/+ 4 e/-4. r-l c« -10 cd- 20 c« + 5 cfo - 20 i c« - 10 1 <^-20 i i i $*. y)' ^1 ^4 :l=10 db20 db20 Jk20 A 10 ^4 :fcl 144] A THIRD MEMOIR UPON QUANTICS. 315 No. Si. No. 35. aeeg + 1 acP -1 ad^g - 1 adef + 2 o^ -1 h'eg -1 h'P +1 bcdg + 2 6ct/^ -2 6dy-2 6d;e» +2 -1 + 2 + 1 -3 + 1 12 a^<P(^ + 1 a^defg— 6 aV/» + 4 aV<7 + 4 aV/> - 3 abcd^— 6 a^«/gr+ 18 a^/» -12 aMy^+ 12 ahde^g-l^ aMf + 6 ao*5^ + 4 ac»c«^ - 24 o^dfg-\% a^eP + 30 ac<Peg + 54 acdy^-12 acrfey-42 ace^ +12 orf*^ -20 ««/'«/• +24 acP^ - 8 W^ + 4 Wg -12 6y» + 8 6V^ - 3 ftW^r +30 6»c«/> - 24 b'iPeg - 12 6«rfy* - 24 6»dey + 60 6V - 27 6cy^ + 6 hi^deg - 42 bccPg bccPef bcd^ bdy d^eg c'def ed^e + 60 -30 + 24 -84 + 66 + 24 -24 + 12 -27 - 8 + 66 - 8 -24 -39 + 36 - 8 :A:665 The seztmvariant may be thus represented by means of a determinant of the dxth order and of the quadrinvariant and quartinvariant. 5xNo. 35 = 4-4(asr-66/+15ce- a, 26, 3c, 4d, e 6, 2c, 3d, 4c, f c, 2d, 36, 4/ 9 a, 46, 3c, 2d, c 6, 4c, 3d, 2c, / c, 4d, 3c, 2/, 9 10*) a, 6, c. d 6, c, d, e c, d, c, f d, c, /, 9 The tables Nos. 36 and 37 relate to a septimia No. 36 is the septimic itself; ^^0. 37 the quartinvariant. No. 86. ( o+l 6 + 7 c + 21 rf+35 « + 35 /+21 9 + f h+1 40—2 316 A THIRD MEMOIB UPON QUAMTICS. [U4 No. 37. aW - 1 6rf«A - 40 abgh ' + 14 bdeg - 50 acjh - 18 bdp - 360 acg* - 24 bf?f + 240 ad^h + 10 <?eg - 360 ad/g + 60 c>P - 81 ae^g - 40 cd^g + 240 b^h - 24 cdef + 990 6y - 25 ce" - 600 bcfg + 234 ciy - 600 bceh + 60 (£«c» + 375 ±2223 The tables Nos. 38 to 45 relate to the octavic. No. 38 is the octavic itse No. 39 the quadrin variant ; Nos. 40, 41 and 42 are the quadricovariants, the last them being the Hessian; No. 43 is the cubinvariant ; No. 44 the quartinvariaut, ai No. 45 the quintinvariant, which is also the catalecticant. No. 38. ( a+ 1 6 + 8 c + 28 (f + 56 « + 70 /+56 ^ + 28 A + 8 » + l W^^yf No. 39. No. 40. ai + 1 bh — 8 eg + 28 df — 56 c« + 35 ±64 ag + \ oA + 2 ai + 1 bi + 2 ci + 1 6/- 6 bg - 10 bh - 2 ch - 10 dh- Q ( cc + 15 cf + 18 eg - S dg + IS eg + lb d« - 10 de - 10 d/ + 34: e/ - 10 /« - 10 e« - 25 ±16 ±20 No. 41. ±35 ±20 ±16 5^, y)^ oe + 1 a/+ 4 ag + Q ah + i ax -^^ 1 6t + 4 ci + 6 c^i + 4 et + 1 6c£ - 4 be -12 bf - S bg + S bh + 12 cA + 8 c^A- 8 eA-12 /A- 4 c« + 3 cd + S. ce -22 cf -48 eg - 22 ety - 48 c^ - 22 fg^ 8 / + 3 cf» + 24 cfe + 36 df - 36 «/ + 36 /' + 24 6^+45 ±4 ±12 ±30 ±48 ±58 ±48 ±30 ±12 $a-, yf No. 42. ac + 1 ad-^r 6 ac+ 15 a/'+20 a^+ 15 aA+ 6 1 ai + 1 6i + 6 ci + 15 rfi+20 «» + 15 /i + 6'j,i + 6'-l 6c -6 6c£+ 6 6c +50 6/+ 90 bg^ 78 6A+ 34 cA+ 78 dh^ 90 cA + 50 /A+ 6 yA-6 A»- / c« -21 ccf-.70 ci« -105 c/ + 126 c^ + 154 cfy+126 />-105 /^-70 ^•-21 ( 1 1 (fo-210 rf/- 14 6» -175 e/ - 210 ±1 ±6 ±21 ±70 ±105 ±210 ±189 ±210 105 ±70 ±21 ±6 14] A THIRD MEMOIR UPON QUANTICS. 317 No. 43. No. 44. aei + I a/h- 4 a^ + 3 bdi - 4: heh + 12 hfy - 8 c»i + 3 cdh - % ceg - 22 cP 4- 24 J^g + 24 (ii'/ - 36 e* +15 ±82 ac<7t - 1 hcgh + 3 cd^i - 2 acA« + 1 hdei + 1 cdeh - 23 adfi + 3 6c(A - 10 c^//*y + 27 ac&/A - 3 hag"" + 9 ce»y + 19 oeH - 2 6eVi + 11 cef - 21 a€/7t + 1 hefg - 23 (^A + 12 alf + 3 hp + 12 (r€g - 21 oPg - 2 c'et + 3 c^/' - 13 6Vi + 1 <?fh + 9 de"/ + 32 6^/i* - 1 c«/ - 12 e* - 10 6r/i - 3 1 ±147 No. 45. 1 i acegi + 1 «/* + 1 bde(f^ 4 CiPg^ + 1 t aceJi^ — 1 I b^egi 1 hiiPg + 2 cdefg - 2 acfi — 1 IPeh' + 1 be>h 2 cdp 2 acfgh + 2 I'M — 2 beVg V 4 c^g 3 acg^ -- 1 ! byi^ + 1 hop - 2 ce^dh ^ 4 (vPql — 1 by + 1 egi - 1 ccT ^ 3 1 (uPh^ + 1 bcdgi + 2 eh^ + 1 dH t- 1 1 1 adefi + 2 bcdli" — 2 <?dfi + 2 d'eh 2 Ofiegh — 2 bcefi — 2 <?d<jh 2 <^fy 2 a<//Vi — 2 1 bceffh + 2 eeH + 1 d'e'g h 3 1 ad/g' -u 2 1 br/Vi + 2 e'efh - 4 it'^P + 3 1 a^i — 1 kf</' — 2 oV + 2 : d^f - 4 1 1 ae^jh T o 1 bdifi — 2 cyv 1- 1 ^ - 1 a^g^ + 1 ' bd'gh + 2 cd^ei - 3 1 aepg 3 1 bd(^i + cd^fh + 2 ±56 If we write i' ■ d 6-12\, No. 39 = /, No. 43== J, No. 44 = ir, No. 45 = £, lambdaic, viz. h . c , d , e-12X ' c , d , C + 3X, / d , e-2\, f 9 e+3\, / . g I f ,9 , h • \ 318 is equal to A THIRD MEMOIR UPON QUANTICS. [144 L + 2\K + 3X»J+ 18V/ - 2592V. Nos. 46 to 48 relate to the nonic. No. 46 is the nonic itself; Nos. 47 and 48 are the two quartinvariants, each of them in its best form, viz. No. 48 does not contain a', and No. 47 does not contain aci^, the leading term of No. 48. The nonic is the lowest quantic with two quartinvariants. No. 46. a+l 6 + 9 c + 36 c^ + 84 e + 126 /+126 ^ + 84 /i + 36 1+9 i+1 No. 47. No. 48. a'/ - 1 b/^h a«/ ... bph + 70 abii + 18 b/g^ - 720 dbij bfg^ - 45 act* cYj + 432 acv" + 2 c^^gf + 27 achj - 72 <?gi - 1728 achj — 2 i^gi - 52 \ adgj + 168 c W adgj + 7 <^h^ + 25 i adhi cdej - 720 orf/a- 7 ccfe; — 45 a^fj - 108 cdfi + 2160 a«^ - 5 cdf% + 23 aegi - 576 cdgh+ 4608 ocgri -22 ce^A + 22 aeh^ + 432 cfh, aefi^ +27 c(H + 70 afH + 540 cefh - 2592 a/«t +25 cefh -127 afgh- 720 eeg" - 5760 afgh -45 c«^ + 32 a^ + 320 c/V + 4320 a^ +20 cpg + 25 6«A; d?j + 320 6*/*/ + 2 cO' + 20 W - 81 d^'ei - 720 W - 2 d^ei - 45 hcgj d'fh- 5760 6c5|[/ - 7 cP/A+ 32 hchi + 648 d«5^ - 1536 tcAt + 7 d'g^ + 47 hdjj - 576 dc/^ + 14688 bdjj - 22 <*!/& + 85 bdgi + 792 de^h + 4320 W^t + 74 d^h + 25 bdh^ - 1728 dp - 8640 bd^i^ - 52 df* - 50 begh + 2160 e»<7 - 8640 ft^^r/* + 23 e*^ - 50 b^j + 540 cy« + 5184 be'j +25 ey« + 30 be/t - 972 6(2/t -73 3[a^ y)*- :k 41650 :A=698 Nos 49, [49 a] and 50 relate to the dodecadic. No. 49 is the dodecadic itself: [No. 49 A, inserted in this place, but originally printed in the Fifth Memoir on Quantics, is the dodecadic quadricovariant]. No. 50 is the cubinvariant. [The numerical coefficients in this last table as originally printed in the Third Memoir were altogether erroneous, and the table as here printed is in fact the table No. 60 j^, of the Fifth Memoir on Quantics.] No. 49. a+l 6 + 12 c + 66 rf+220 c + 495 /+792 ^ + 924 A + 792 i+495 J +220 A;+66 f+12 m + l 5«,yr 144] A THIRD MEMOIR UPON QU ANTICS. 319 16 No. 49 A. ag^ 1 ah+ 6 at + 15 a;+ 20 ak-i- 15 6/- 6 6<7-30 bh- 54 bi- 30 6; + 30 c« + 15 C/ + 54 eg + 24 cA - 150 ci -270 ( d'-lO cfo-30 rf/+150 dg + 430 ^+270 e« - 135 ./ - 270 c^ +495 /2 - 540 aZ + 6 bk+ 54 c; - 150 di- 270 «A + 1080 fg- 720 am + 1 bl + 30 cA; + 24 dj -430 c» +495 //i +720 ^ -840 =1:60 :l=189 :t460 rtSlO :tll40 ±1270 brn + 6 c/ + 54 dk - 150 ei - 270 /» +1080 gh - 720 cm + 15 dl + 30 cifc -270 ^ +270 ^i +495 h^ -540 c/m + 20 el - 30 />fc -150 <d +430 Xi -270 cm + 15 /I - 54 i^ifc + 24 hj +150 -135 t« /m + 6 ^/ -30 A>fe +54 y -30 gm + I \ A/ - 6 ' tA; + 15 ty- \n j2 ^10 i$*»y)" ±1140 ±810 ±450 ±189 ±60 ±16 No. 50. agrn + 1 c/l - 54 dhi + 270 oA/ - 6 c^A; + 24 e^A; -135 aik + 15 chj +150 ^Jj +270 a/ - 10 ci« - 135 c^i + 495 bfm — 6 <Pm - 10 eh* -540 6^; + 30 rfe^ + 30 /S - 540 bhk -54 dfk +150 fgh + 720 6v + 30 ^^- _430 ^ -280 cem + 15 ±2200 Resuming now the general subject, — 54. The simplest covariant of a system of quantics of the form (where the number of quantics is equal to the number of the facients of each quantic) is the functional determinant or Jacobian, viz. the determinant formed with the differential coeflScients or derived functions of the quantics with respect to the several facients. 65. In the particular case in which the quantics are the differential coefficient8 or derived functions of a single quantic, we have a corresponding covariant of the single quantic, which covsuriant is termed the Hessian ; in other words, the Hessian is the determinant formed vrilth the second differential coeflScients or derived functions of the quantic with respect to the several facients. 66. The expression, an adjoint linear form, is used to denote a linear function fe + i7y+ .... or in the notation of quantics (f, i;,...$a?, y,...), having the same facients as 320 A THIRD MEMOIR UPON QUANTICS. [144 the quantic or quantics to which it belongs, and with indeterminate coefficients (f, i;,...)- The invariants of a quantic or quantics, and of an adjoint linear form, may be considered as quantics having (f, 17,...) for facients, and of which the coefficients are of course functions of the coefficients of the given quantic or quantics. An invariant of the class in question is termed a contravariant of the quantic or quantics. The idea of a contravariant is due to Mr Sylvester. In the theory of binary quantics, it is hardly necessary to consider the eontra- variants; for any contravariant is at once turned into an invariant by writing (y, — d?) for (f, 17). 57. If we imagine, as before, a system of quantics of the form (•$^, y, ...)'«, where the number of quantics is equal to the number of the facients in each quantic, the function of the coefficients, which, equated to zero, expresses the result of the elimination of the facients from the equations obtained by putting each of the quantics equal to zero, is said to be the Resultant of the system of quantics. The resultant is an invariant of the system of quantics. And in the particular case in which the quantics are the differential coefficients, or derived functions of a single quantic with respect to the several facients, the resultant in question is termed the Discriminant of the single quantic; the discriminant is of course an invariant of the single quantic. 58. Imagine two quantics, and form the equations which express that the differen- tial coefficients, or derived functions of the one quantic with respect to the several facients, are proportional to those of the other quantic. Join to these the equations obtained by equating each of the quantics to zero; we have a system of equations, one of which is contained in the others, and from which therefore the facients may be eliminated. The function which, equated to zero, expresses the result of the elimination is an invariant which (from its geometrical signification) might be termed the Tactinvariant of the two quantics, but I do not at present propose to consider this invariant except in the particular case where the system consists of a given quantic and of an adjoint linear form. In this case the tactinvariant is a contravariant of the given quantic, viz. the contravariant termed the Reciprocant. 59. Consider now a quantic (•Ja:, y,...)*", and let the facients x, y, ... be replaced by Xx + fiX, Xy-h/^F, ... the resulting function may, it is clear, be considered as a quantic with the facients (\ fi) and of the form 144] A THIRD MEMOIR UPON QUANTICfS. 321 The coefficients of this quantic are termed Emanants, viz., excluding the first coefficient, which is the quantic itself (but which might be termed the 0-th emanant), the other coefficients are the first, second, and last or ultimate emanants. The ultimate emanant is, it is clear, nothing else than the quantic itself, with (X, F, ...) instead of (x, y, ...) for facients : the penultimate emanant is, in like manner, obtained from the first emanant by interchanging (ar, y, ...) with (X, F, ...), and similarly for the other emanants. The facients (X, F, ...) may be termed the facients of emanation, or simply the new founents. The theory of emanation might be presented in a more general form by employing two or more sets of emanating fii<;ients; we might, for example, write \x-\-fi,X'{-vX\ \y •{■ fiY + vY\ . , , for x, y, ..., but it is not necessary to dwell upon this at present. The invariants, in respect to the new facients, of any emanant or emanants of a quantic (i.e. the invariants of the emanant or emanants, considered as a function or functions of the new facients), are, it is easy to see, covariants of the original quantic, and it is in many cases convenient to define a covariant in this manner; thus the Hessian is the discriminant of the second or quadric emanant of the quantic. 60. If we consider a quantic (a, 6,...$a?, y,...)'^, and an adjoint linear form, the operative quantic (which is, so to speak, a coutravariant operator) is termed the Evector. The proper- ties of the evector have been considered in the introductory memoir, and it has been in effect shown that the evector operating upon an invariant, or more generally upon a contravariant, gives rise to a coutravariant. Any such coutravariant, or rather such contravariant considered as so generated, is termed an Evectant In the case of a binary quantic, (a , b ,...$ar, y)"», the covariant operator {da, 96,...$y,-^)"* may, if not with perfect accuracy, yet without risk of ambiguity, be termed the Evector, and a covariant obtained by operating with it upon an invariant or covariant, or rather such covariant considered as so generated, may in like manner be termed an Mvectant, 61. Imagine two or more quantics of the same order, (a, 6, ...][a?, yf, (a, /8,...$a?, yf, -we may have covariants such that for the coefficients of each pair of quantics the covariant is reduced to zero by the operators ad^ + Idp 4 . . . , ad a + I3di, -h . . . . C. II. 41 322 A THIRD MEMOIR UPON QUANTICS. [l44 Such covariants are called CtmbinanUy and they possess the property of being inva- riantive, quoad the system, i. e. the covariant remains unaltered to a factor priSy when each quantic is replaced by a linear function of all the quantics. This extremely important theory is due to Mr Sylvester. Proceeding now to the theory of ternary quadrics and cubics, — First for a ternary quadric, we have the following tables: — Covariant and other Tables, Nos. 51 to 56 (a ternary quadric). No. 51. The quadric is represented by which means cwj» 4- 6y' + c^ + %fyz + igzx 4- 2Jucy . No. 52. The first derived functions (omitting the factor 2) are — (a, A, gjx, y, t), (A, 6, fjx, y, z\ No. 53. The operators which reduce a covariant to zero are ( h, 6, 2/ $3^, df, dc)'-zdy, ( a, 2A, g^Ky 9b, d/)-ydg, ( g, 2/ c$3a, 3*, 3/)-y3„ ( a, A, 2g'$dg, 3/, 3c) --^3,, (2A, b, f^da, 3a, 3^)-a3y. No. 54. The evector is (3a. 3ft, 3c, 3/. dgy dk\^, 17, f)'. 144] A THIRD MEMOIR UPON QUANTICS. 323 The discriminant is which is equal to No. 55. a, h, g h, b, f abo - af* - b^ - ch* + 2fgh. No. 56. The reciprocant is f, a, A, g Vf K b, f which is equal to (6c-/», ca-g\ ah-h^ gh-^af, hf^bg, fg - ch\l ri, ^. The discriminant is, it will be noticed, the same function as the Hessian. The reci- procant is the evectant of the discriminant. The covariants are the quadric itself and the discriminant; the reciprocant is the only contravariant. Next, for a ternary cubic, we have the following Tables : Covariant and other Tables, Nos. 57 to 70 (a ternary cubic). No. 57. The cubic \a U^ (a, 6, c, /, g, A, i, j, k, Z$a?, y, zf, which means — aa^ + bf-¥c2^ + Sfy^z + Sgz^x + Sha^ 4- %-&• + Bjza^ + 3fcry" + 6lxyz, No. 58. The first derived functions (omitting the factor S) are (a, k, g, I, j, hjx, y, z)\ {K b, %, f, I, kjx, y, zf, (jy /, c , i, g, I '$x, y, z)\ 41—2 324 A THIRD MEMOIR UPON QUANTICS. [144 The second derived functions (omitting the factor 6) are (a. h. j $a;, y> ^). (k. b. fl^. y> A iff. • c l^x, y. A (I. /. ijix, y> '). a I. gl'^' y> '). (h, k. I T§ix, y. ')■ No. 59. The operators which reduce a covariant to zero are (j. 3/. c, 2t, 9. (o. k. 3«7. ii, 2j. (3A. b, • / 2«, (A. b, 3i-. 2/. 2i. (3j, /. c, • t, 2*7. ( a, Sk, S'. 21, i. The evector is 2Z$3a, 96, 9<, 9/, 9(, A$9j, 9/, 9c, 9,-, dg, 2kJida, 9*, 9^ ff* 9^ 9j, dk) ^][3i> ^/» 3c> 3»» ^y» 90 — '2r9y, 2Z$9a, 9*, 9^, 9f, 9,-, 9*) — a:9,, 2A][9a, 9^, 9<, 9/, 9^ 9*)-y9«. No. 60. (9a, 9^, 9c, 9/, 9y, 9a, 9<, 9;, 9*, 9f3[f, 17, (;)*. The Hessian \a HU^ (a, A, j$ir, (A, A;, Z $ar, which is equal to No. 61. y, 2:), (A, A;, Z$ir, y, ^), (j, ^» 3^$^* y, ^), (Ar, 6, /$a:, y, ^), (Z, / i$a:, y, ^), (i, /, t$a?, y, 4 (jr, t, c$a?. y. ^) y. ^) y. z) ' agk-l Mi-1 cy -1 6<*-l ae/-\ abg- 1 bcj -1 «cA — 1 aW -1 06c — 1 aP +1 6P +1 c? +1 65-^+2 ot* +1 flt/*/ +2 bf +1 «/ff-l oT + l q/i +1 yA' +1 /»A+1 fg" +1 6»y -1 cAZ +2 oii — 1 c/l-1 ail +2 6yA-l hgj +1 A/i -2 /«-2 fki -2 c*» +1 # -1 bp +1 c^/ +2 eh" + 1 y/ +2 cAA + 1 j»* +1 ♦*• +1 iy +1 /y +1 /»■ +1 /Hi -2 yv +1 ^ +1 ^k-2 /yA-3 •\ /y*-2 g'k +1 J7W+I j;i*-2 g;A + 1 gJ^ +1 fjl +2 y%i + i j^At-2 AS +1 At» +1 hij -2 AK+1 yA/+2 fp -1 j?P -1 A^ -1 iP -1 iP -1 AP -1 Aa +2 V* -3 v. P -2 <l(«,y, :i=3 :i=3 dbS =lr5 ±6 :t5 ±9 144] A THIRD MEMOm UPON QUANTICS. 325 The quartinTariant w S= No. 62. abcl — 1 /y +1 abgi + 1 fghl + 3 acfk + 1 /a;*-i qTg^l /Av -1 afil + 1 ^? -2 aPk -1 i^ifc« +1 hch^ + 1 ghki — 1 hfh -\ ^A;? -2 bgjl + 1 AV +1 hij" -1 A*? -2 qf'h -1 ijkl +3 cM/ + 1 ^* +1 ciife« -1 il6 The sextiu variant is 7 = No. 63. a^¥<^ + 1 acpKL -24 hcfhj^ -12 cfhH - 12 fgikV^ -12 a^bcfi - 6 acpjk -12 hcghH -24 cfh^J^ + 12 fh^3 -12 a^hi? + 4 acfg]^ -12 hcghjk + 18 cfhjJd 60 fhijl? -12 a^cP + 4 acfJiki + 18 bchHj -12 cjr^ + 24 fim + 36 a^PP - 3 cicfkJ^ + 36 hchjt' + 36 cgWl + 12 fU^ + 24 ab^cgj - 6 acH^l -24 hcPkl -24 cgjl^ -12 9"^ + 8 ahy + 4 ^Poi -12 hfg^hj + 6 chHkl + 12 g'hkH -12 ahc'hk - 6 afVk + 24 Pf9?l + 12 chijh^ + 6 ipJ^P -24 ahcfgh + 6 apghi + 6 hfif -12 chkU" -12 gh^ki^ -12 abcjjl + 12 cpgJ^ + 12 hg'hk -12 i^iei^ + 12 gJJdP -12 ahcgkl + 12 a/Hjl + 12 hg'hH + 24 fV + 8 gij^l + 36 abchil + 12 a/gkil -60 hfh^ + 12 /yA« -27 gkl* + 24 abcijk + 6 afhi?l + 12 hfikl + 12 /v* -12 h^i? + 8 abcP -20 af?ik + 6 hghijl -60 Pghji + 36 hS*J^ -24 ah/gij + 18 afU? -12 bgipk + 6 phis' -12 hit" + 24 ohffl -24 agJ^il' + 24 bgjP -12 yy^ -24 hi^kl + 36 abg'ki -12 aht^k -12 bhPf + 24 ffhki + 36 tVifc» -27 ahghi^ -12 ai»kP + 12 bipP + 12 ffi^ -12 ijkP -36 abgU^ + 36 ^cP + 4 (?h^l^ - 3 fghHl + 36 Z» - 8 ahi^jl -24 ^9T - 3 cPh^j + 24 fghijk - 6 04?}^ + 4 h(?h^ + 4 cfgh^k + 6 fgh^ -36 ±871 The discovery of the invariants S and ressions were first obtained by Mr Salmon. T is due to Aronhold, the developed No. 64. There is an octicovariant for which we may take eu= d^u, dyHU, d^Hu, fa,' u, id^dyU, dyHU, idyd^u, J 3/ u, dHU, ifd.d^u, idyd,u. d,HU 326 A THIRD MEMOIR UPON QUANTICS. [144 or else ©.tr= ia, u, ^dy u, id, u id^U, a,» HU, d,dyHU, d^,HU j IdyU, dyd^HU, dy* HU, dJd^HU J a, £7, d,d,HU, d.dyHU, a,» hu or else, what I believe is more simple, a function 9„U, which is a linear function of the last-mentioned two functions. The relations between SU, S,U, S^U are -S,U+4&U=T. V^-2iS.U.HU, e„U+2SU=T.U*-108.U.H(T. I have not worked out the developed expressions. The cubicontravariant is PU= No. 65. / 1 bcl -1 ad — 1 ahl^l aek + 1 o6i + 1 ftc; +1 oiy + 1 bch+l a^ + 1 abc-l 6^+1 a^ + 1 a/k+l afg-2 «/"-! V-1 a/if +1 bgl +1 ot* -1 <j/» + 1 c/k + l chj + 1 hhj + 1 ail + 1 6j^A-2 c/A-2 atA - 2 6V -2 eU. +1 *»■ +1 Pg-\ g'h^X /A>-1 ch* -1 6;7 +1 ckl +1 y« -1 cA» -1 <2;* -2 cM + l • fil +1 gjl +1 cM +1 /jf +2 /hi + 3 /i^i +3 yv-i /V + 2 /a> -1 /gh+3 I'A: -1 ip -1 j^ -1 //*/ + 3 j!;*-i yv -1 ghk-l /?*-! g'k+2 fjl -4 ajk -1 S-A* +2 gki-l hH +2 /A»-l ghi-\ y«-4 hij - 1 AH-1 hi? +2 hP -2 /P -2 gP -2 Atf -4 jP -2 itP -2 t? -2 i« +3 Ud +3 ijl +3 V* +3 V. P +4 ±3 ±3 :l=8 ±7 ±7 ±7 ±7 :fel8 <[^, 17, 0'. 144] A THIRD MEMOIR UPON QUANTIC8. 327 No, 66. The quintic contraTariant is QfT^ + 1 a«6<r» + 1 a«6V + 1 a'^ftci - 3 a6 V - 3 o^c»A:- 3 a^bcf- 3 o^'c^- 3 aftc'A- 3 cibcfj + 6 - 6 a^cfi- 3 a»6/» - 3 aV + 6 aby + 6 ahc/g+ 3 a'^ftt' + 6 abcfl+ 6 rt^c^^ + 6 oAc^A + 6 + 4 aH** + 2 ay + 2 ayi«- 3 abc/h+ 3 a6cfi + 6 a^fH- 3 o^ctA; + 3 abcij + 3 ahchi + 6 + 4 ohcg^ — 6 ai^si- 3 a6(^A+ 3 a6cA;^ + 6 abgi^ — 6 a6cA7 + 6 ahfgi + 9 abfi- 6 ah<^ -30 - 3 ab^ + 4 a6cM— 6 a6c;7 + 6 ab/ij+ 9 ac/«Z - 12 ahcjk + 3 aAt«/ -12 ac«Ar» + 6 oW - 12 - 3 at?hk- 3 a6/<7^+ 3 ahgii + 9 ab/gl-24: acfki+ 9 afc/k; + 9 acpk- 6 acpj- 6 ahgU + 36 + 2 acfgh+ 3 ab/jl+ 6 ab^l -12 abgki— 12 a/V+ 3 aft^A; - 6 a/V - 6 ac/^A;- 12 o^i^J - 12 - 3 acfjl + 6 abgkl+ 6 ac/A^-24 o^/a^- 6 a/iH + 6 ahqhi— 12 a/*t/ + 6 oc/'Ai + 9 acfVi-\2 + 3 acgkl + 6 a6Ai^+ 6 ac^A;-12 a6iZ« +18 ai'A; - 6 06^/^ + 18 a/*i^A; + 3 ac/P + 18 acfkl + 36 + 6 achU + 6 o^vA; + 3 ctcgl^— 6 ac/A:»- 6 bc'h^ + 6 aMil - 24 6 V + 6 ociAr^ - 24 acih? - 12 + 6 acijk + 3 ahP -10 acAA:t+ 9 a/^j - 6 bcff^ - 6 acfhk+ 9 bYJ - 3 a/V + 12 a/V/ + 12 + 6 acJ^ -10 acil» + 4 oc*;* + 18 a/VA;+ 24 ici^A^ - 24 acA:V - 12 be/ hj- 12 afgil - 30 aAy+ 6 + 3 «/&V + 9 a/«A/-12 af^gj-l^ a/^hi+ 3 bcgjk + 9 a/VA+ 3 bcghk+ 9 a/i«J + 3 afgki - 30 -10 affl - 12 a/y*- 6 afg'k + 24 o/«P + 6 6c/«>*-12 «/!;7+ 6 &cA«t - 6 agki^ + 24 a/At«+ 6 + 9 a^ki — 6 afyk^- 6 afghi + 6 qfkil - 30 bcjP +18 o/^A;; - 30 6cAP + 18 aAt' - 6 afiP - 18 -12 (ighi^ — 6 a/hki+ 9 o/i^P + 12 oAr^i^ + 12 */i^!; + 3 a/hil+ 12 bcjkl - 24 a?P + 6 ai'A/ + 12 - 6 agi^ +18 o^^P + 18 afijl + 12 l^gP - 3 bg'k - 6 fl/t; A; + 6 ft/^A+ 3 bcghj + 9 bcgh^ - 12 - 6 ai?jl -12 o*A:»; - 12 flw/^iY - 30 bchH - 12 6/Ai + 24 a/P - 6 Vk;'^ + 12 6c// -12 bchjl + 36 + 18 6cf + 4 ^/ + 2 ahn + 6 fccA/A: + 9 V^ + 6 a^A:^ +24 b/if - 18 bg% - 6 6c/A - 12 12 V/ - 3 6cA» + 4 ai^'A; + 3 b/g/ij+ 6 6^^; - 30 aAi2Ar-18 ^^iT'A:/ + 6 Vi^ + 6 bfgp + 6 + 2 c»A» + 2 b/hj'- 6 aiP - 6 ftj5^^ + 6 biY +12 aikP + 12 bghil - 30 bgij' + 3 6i^A/ + 12 -12 cfhp- 6 6^A»^-12 bchj' - 6 bg^hk- IS c«AAr» - 3 6cAV - 6 6^/;A: + 6 (T^A^A - 3 bg^jk + 6 - 6 c^A«/ - 12 bghjk + 9 6^Ai+ 3 bghH + 24 cj^hj + 24 b/f - 6 bgP - 6 c/gh^+ 3 bghij - 30 - 6 cghjk-k- 9 &A«t; - 6 b^l + 6 bghP + 12 c/ghk+ 6 6(/W + 12 bhi^j +24 r/^i^ - 30 6^'/« -18 + 9 c/Ay — 6 bhjP + 18 6v» - 6 6fiyAr; + 12 c/A^ - 18 6^A;7 - 30 6i;P +12 c/j'k + 24 bifl +12 + 18 chjl^ +18 6/)W - 12 cm + 24 bhijl - 30 c/hP + 12 bgfk + 3 c/Vi'^ + 12 c^AA;+ 12 c/A«/ + 12 -12 c/A:Z -12 c/i^k^ - 3 c^A«A;+ 3 6i/«>fc + 3 ciO'A;^ - 30 6Aty« + 24 c/hkl - 30 cgjk^ - 18 c/A;A - 30 - 6 //Ai+ 3 /»/i!/ + 12 chH - 6 bjP - 6 cg^l + 6 b/P + 6 c/j/(^ + 24 chHl + 6 cgrAA* + 6 + 12 A;"'^ + 6 /^/*»A:+ 3 cA«Z» + 6 c/h^k+ 3 cAiAr^ + 12 r/A» - 6 c^A:* - 6 cAyA + 6 chHk + 6 + 3 fif - 6 /hH - 6 chjkl - 30 cAA:*/ + 6 cij^ + 3 cA«A:; + 6 cAiA:« + 3 cAZ» - 6 chkP - 18 + 6 ^iW: - 6 /A«^ + 6 cf^ +12 cjk" - 6 ckP - 6 chjk" + 3 ck'P + 6 9 A/2 +12 cjm +12 + 6 g'hH +12 fhjkl-ZO /y +12 /V*' - 27 /yA - 27 /V- 6 /T +12 / V - 6 /V^i + 18 -30 i^AZ« + 6 ffl^ +12 //A« - 27 /!;"'^ - 6 Pgil + 18 /^A«/ + 18 PiUk -12 /^A/ + 18 Pjn -24 + 6 g'jkl + 6 i^AJfe«Z+ 6 /«;•«* -12 /%7 + 18 Z*^/* - 6 /^A;*- 3 Pghl + 18 fg'jk -12 /</»M + 18 + 3 ^Av7 - 30 9i1^ - 6 /^/i;7 + 36 /ghkl + 36 /v'*^ + 18 /hHj -12 /Va>- - 12 /^Ai;- 3 fghH + 18 - 6 gipk + 3 A«tA:Z + 6 /hip - 12 /^•^-12 fqhil + 36 /Ai^ - 6 PP -24 /^•^ - 6 fghP - 54 + 12 <Z;7» - 6 Ai/jfe« + 3 ;!rP -24 /A«i/ + 18 /W^- 3 ffkl + 18 //A? - 6 /(;•'/ + 18 /7iA/ - 12 - 6 h%y +12 MP - 6 ^hkl + 18 /Ai/A- 3 fgl? -18 i^AA:> - 6 fghik-Z g'k' +12 /Ai;7 -12 + 6 i7/» + 6 iA«P + 6 !/!;^ - 6 /A/» -18 yAt^i -12 gh^ki- 12 fgkP - 6 g%ki-\2 /i/*A+18 ghHl + 18 j5^p - 6 /i/P - 6 ghkP - 6 fhS" - 6 fkP -24 /;/» +48 ghijk - 3 ^F +12 fiei - 6 gjm + 18 /AiP - 6 ^A«i» - 6 fkH -24 ^/i^» -18 (^/lA^^i - 12 ghk^ - 12 A\^ + 12 fijkl + 36 ghiP - 6 ghkil- 12 i^'/t/^ - 6 gk^P -24 i^>ti7» - 6 hHP -24 /^ + 12 ^(/A/ + 36 gij^ + 18 AVy - 6 /i«A:i« - 6 AV + 12 A/^ + 12 giMl + 18 gP + 12 ^A/* +48 hip - 6 fikiP - 6 Ai«P -24 hijkl + 36 Ai^A;; + 18 At^/ +18 AW -24 V*A;Z +18 t/A:*/ +18 i^jkl +18 ifJ^ -27 t^A* -27 i^j'k -27 hiP +48 p + 12 kl* + 12 t7* + 12 JAjP -18 ikP -18 iA/» - 18 Ai!/A + 18 ijkP -54 Z» -24 '\ ^{t V (r dbl45 14o i:282 =b282 :i^282 :1=282 :t282 282 db486 328 A THIRD MEMOIR UPON QUANTICS, [144 No. 67. The reciprocant is FU={*^^, 17, f)' = ^ W +1 bc/i - 6 W» +4 cP +4 aV +1 acgj— 6 o^ +4 cf +4 pV-3 f^. a»6» +1 abhk — 6 oifc* +4 6A» +4 h^Jc" -3 ^r. r»^ 9 * - 6 aegh + 6 ag'l chf if + 12 + 18 -24 -12 + 6 + 12 -12 abfh ahkl afie hhH bhjk fh^k hiei - 6 + 6 + 12 -12 -24 + 18 + 6 + 12 -12 iS- hi?k hcfg bcil qfH - 6 + 6 + 12 -12 -24 c/H + 18 ft'l +12 « - 12 iji*. a V - 6 abhl + 12 abjk + 6 a/hk + 18 ak'l bhy -24 -12 -12 + 12 + 6 if. l^cg btifl bcik hfgi bH'l cPk P9 A^k - 6 + 12 + 6 + 18 -24 -12 -12 + 12 + 6 fif". ac»A - 6 ciegl + 12 acij + 6 agH -12 cghj + 18 cpi -24 ^A -12 9^1 +12 ^i" + 6 ±9 9 9 54 54 54 54 db54 54 n*i*. W W. i?»f*. i*e*. ev^> T'f. f*?. ?»i». a V + 6 o^V + 6 6c«A + 6 a»6» + 6 6»c/ + 6 ac'k + 6 a'be - 2 a6«c - 2 oW - 2 aV + 9 ab/l - 12 6c^/ -12 ay + 9 by + 9 acfg - 6 aVi - 18 a6/t + 6 ac/» + 6 achl - 12 abik — 6 bcij - 6 a5^/* — 6 fcc/A - 6 acil -12 abgj + 6 a/» - 4 at' - 4 oc; A; - 6 a/»A; + 12 6^ +12 abjl -12 6dW -12 agp + 12 achk + 6 6«a; -18 ftca; + 6 a/gj -18 ^i" + 9 c»A:» + 9 a/A/ - 36 b/gl -36 c»A« + 9 afgh + 18 6cA)k+ 6 V - 4 o^A; +12 6/A; - 18 c/y +12 a/jk - 18 b/ij -18 cJP + 12 afjl +36 bfgh + 18 c'Ait -18 aghi - 18 bghk - 18 (2/5^A; - 18 agk" +12 6^ -18 cghl -36 agkl - 48 ^^gf/ +36 c/flrA + 18 a^^ +48 bhH +12 c/hi - 18 atAifc - 18 6Ai» +12 cgjk -18 ahU +36 bgld + 36 cfjl -48 aijl -36 6AZ« +48 c//» +48 oA^ +48 biP +iS chij -18 aijk +18 6AiZ - 48 <^A^ +36 cAV +12 6;A^ -36 o^Z -36 bkp +12 c/*» +12 cjP + 48 a/* -32 bijk +18 cijk +18 or +12 /W - 3 /y - 3 ^A'A: - 6 /!/ +12 m - 6 6;' - 4 6/" -32 cAt/ +36 i^A» - 3 /AA;Z - 24 /gil -24 A'j +36 /^gk + 36 fk + 12 cA» - 4 cP - 4 c/» -32 5rA;7 -24 ^k" +36 /»!; - 6 A«t +12 /•Af - 6 fhi + 36 /y -36 /»AZ + 12 /^Z +12 Sfk - 6 ^/fc* +12 gi'k +36 h^P -12 PP -12 g'P -12 ghH +12 Pjk - 36 ^V +12 Atf +36 hik" - 6 Ai» + 12 A;;M -24 /a/ - 24 gijl -24 (^Ajife + 12 /i7ife» -36 g'ki -36 fP -12 *»Z» -12 *•/« -12 /*» - 3 Pk" - 3 »y - 3 AV -36 hjP +24 y»« +12 fhki +12 y^ +24 tib"Z +12 ^At« -36 ^ +24 Pjl +12 135 135 :i^l35 135 dbl85 185 180 11x180 rfelSO U] A THIRD MEMOIR UPON QUANTIC8. 329 12 30 12 24 18 12 24 24 12 18 60 12 66 12 48 60 V*^' ac/j acgk achi agtl ai?j egh^ chjl cpk fhl fjk ghij jHl -12 -12 + 30 -24 + 24 + 12 -18 -18 + 12 + 24 -12 + 60 -12 -66 -48 + 60 i^fi?. abgk abhi abij abP aph afkl aiJ^ hgh^ hhjl bj^k fhH fhjk gW h^ki hkP -12 -12 + 30 -24 -18 + 12 + 24 + 24 + 12 -18 + 60 -66 -12 -12 -48 + 60 wr. cibci aep a/t^ hcgh hcjl hfl igij cfhl cfjk cg^ chik ckP fgh% gikl hn i^jk + 6 -12 + 6 -24 + 36 + 12 -30 -60 + 12 -36 + 54 + 24 -24 + 60 + 78 -96 + 72 -12 -48 -66 + 48 7»r»^ abcQ abg^ acfh ackl afgl a/ij ahi^ agik ail^ chH chjk fghj m g'hk ghH ghJI? gjkl hijl ij^k + 6 -12 -24 + 36 -60 + 54 -36 + 12 + 24 + 6 + 12 -30 + 78 -48 -24 + 60 -96 + 72 -12 -66 + 48 f*ft7». abck ah/g Ml api a/ik hch^ hghl bgjk bhij bp cm fghJc fhH fhJ^ fi^i gm hikl + 6 -24 + 36 + 12 -30 -12 -36 -60 + 54 + 12 + 24 + 6 + 60 + 78 -24 -96 -12 -48 + 72 -66 + 48 ^17^'. abcf abi^ afH bchl bcjk bfk bghi bffP bijl e/hk cm Ml fhil fijk fP giJ^ hi^k ikP + 6 -12 + 6 + 36 -24 + 54 -36 + 12 + 24 -60 -30 + 12 -66 -48 -12 + 72 + 78 + 48 + 60 -24 -96 iy»rt«. ahcg acfl acik ofgi aiH bcp Vi cghJc chH chP cjkl fail g'kl ghil gijk hiy + 6 + 36 -24 -30 + 12 -12 + 6 + 12 + 54 -36 + 24 -60 -66 + 72 -24 -48 -12 + 78 + 48 + 60 -96 f'^^. ahch abgl abij a/gk afhi afP aikl hghj bfl ch^k fhjl m ghkl h^il hijk hP jkP + 6 + 36 -24 -12 -36 + 12 + 54 + 24 -60 -30 + 12 + 6 -66 -12 + 60 + 72 -24 -48 + 78 + 48 -96 wr». abcl aJbgi acfk oPg afxl w?k bchj bgVi bgjl bip cfli^ chkl \fgh.l /jp g'i^ ghkl gkP h^? hiP ijkl I*. }22 db222 =1=222 :i=408 :Jb408 ±498 ±408 408 408 24 6 6 30 48 30 6 30 48 30 30 48 30 24 + 108 -114 - 114 + 24 + 24 -114 + 24 + 24 + 24 + 108 - 48 ±558 + + + + + + + + + + The preceding Tables contain the complete system [not so] of the covariants and ntmvariants of the ternary cubic, i.e. the covariants are the cubic itself U, the lartinvariant 8, the sextinvariant T, the Hessian HU, and an octicovariant, say SU; e contravariants are the cubicontravariant PU, the quinticontravariant QUy and e reciprocant FU. The contravariants are all of them evectants, viz. PU is the evectant o{ 8, QU the evectant of T, and the reciprocant FU is the evectant of QU, or what is the me thing, the second evectant of T. The discriminant is a rational and integral function of the two invariants; repre- Qting it by E, we have i2 = 64 8^-1^. If we combine U and HU by arbitrary multipliers, say a and 6)8, so as to form e sum aU +6/3HU, this is a cubic, and the question arises, to find the covariants d contravariants of this cubic : the results are given in the following Table : 7-^6^HU {aU-^6/3HU) aU-^e^HU. (0, 2S, T, + (1, 0,-125, No. 68. 8S«3[a, /3yu C. II. 42 aso A THIRD MEMOIR UPON QUANTIOS. 144 P(aU+6^HU)= (1. 0, 125, iT^a, ^yPU 4(0,1. 0, - 'iSlia. ^y QU. (0, 60.Sf, SOT, 0,-1 20T8. - 242* + 5765* \a, fiy P U + (1, 0, 0, lor, 2405*, 2^T8\a, ^yqu. (5, T, 245', 4^5, 2*-485»3^a, ^y. {T, 965", 6075, 202*. 2402*5*. - 482*5 + 46085*. - 82* + 5762'S»'J[a, jS/. [(1,0, -245,-82', -4&S'\t, ^y\*R. (1,0. -245, -87, - 4»S''$_a, 0y FU + (0. 24, 0, 0, -isT^cL, fiy(PUy + (0. 0. 24, 0, 9QS\a, fiyPU.QU +(0, 0, 0. 8. 03[a, fiy.{QUy. We have, in like manner, for the covariants and contravariants of the cubic GttPU+l3QU, the following Table: 5 (aU + 6fiHU) = T(aU+6fiHU) = R(aU+Q^HU) = F(aU+6fiHU) = No. 69. 6(xPU+fiQU =fi<iPU + PQU. H(6aPU + fiQU) = (-2T. 485*, 182'5, 2* + 165»'5a, ^yPU + (85, T, - 85«, - T8 \<i, ^y Q U. P (6aPU+ ffQU) = (325*, 122'5, 2* + 325', 42'5'3^a, /9)» U + (47, 965'. 122'5, 2* - 325* 3[«, ^S)* HU. Q(QaPU + 0QU)= r + 3842'5», + 1202*5 + 7680 5*, + 102* +320075*, + 4802*5», + 302*5, 1^+ 12^ - 242*5* + 6125" + (- 242* +4608 5>, "^ + 19202* 5*. + 4802*5, + 302'* + 19202'5», + 1202"5» + 7680 5*, 62"5 + 76875* 6 a, ^yU ) a, fiy HU. 44] A THIKD MEMOIR UPON QUANTIC8. 331 8(6aPU + ^QU) = T(6aPU+l3QU) = (+ ir» +192 S", ' + 128TS\ + 18T*S + 384 S\ k a, fi)\ + IT* + 64^/8", t+ 5r»S«- 64 -Sf" J '- 8r» + 4608rS», + 1920r»iS» + 73728 /S*. "i + 360r»<Sf + 384002'/S', + 2or* + sgeor'/S", + 84or'/S»+ lesoTS', + S6T*S + SSiPS* + 24576 S', {+ 12" - 40r'/S»+ 2560rS* ; + /3QL0 = [(48&', Sr, -965*, - 247/8, - 2" - 16/Sf "go, /3)']»i?. f/8Qfr) = ( 192S, 322* ,-384 S' ,- 96T8 + ( , 512 iS», 1922* /Sf , 24r'/Sr + (1344S', 3522'«S, 24r'-1152S», -2882'S» + (48 2'. , 2882' -S , 247' + 1536/8*, 6 a, /3)'. 47' -64fi^3[a, /3)*eir 2'' 5a, /9y. U* 20T'8 + 64/S*3ia, /9)« U. HU The tables for the ternary cubic become much more simple if we suppose that 16 cubic is expressed in Hesse's canonical form; we have then the following ible: U S T R HU SU e.u = No. 70. ^ + y* + •** + Qlxyt. -l+l*. 1 _ 20? - 81*. - (1 + 8P)». I* (x* + y* + z*) - (1 + 21') xyz. a + 8J*y {fi> + z'a^+ ai'y*) + (-9l*)(x' + y* + ^y + {-2l- ol* - 20r) (a;» + y» + ^») xyz + (- 151* - 781' + 121*) ai'y'z*. 4 (1 + 8l*y (y*!* + 1^*0^ + a?y*) + (-l-W-4>l*)(ai> + y* + z»y + (4/ + lOOi* + 112P) (a!» + y» + «») xyz + (48f + 552P + 48f) !d*y*z*. 42—2 332 A THIRD MEMOm UPON QUANTICS. [144 ^,,U= - 2 (1 + Sl'fifz' + 2^0^-^- a^f) + (i-ioz»)(^ + y' + -8*)» + (6i - 180Z* + 96^) (a^ + y» + ^) ayyz + (6f» - 624P - 192i«) a^fz\ PU =-Z(p + i?* + (r) + (-l + 4?)^7C. QU =(l-10P)(f* + if + (r)-6?(5 + 4i»)fi;?. FU = - 4 (1 + 8P)(i;»?» + ?»p + fi7») - 24Z (1 -f 2f») fi7»^, to which it is proper to join the following transformed expressions for ©CT, ©,i/, B^^IT, viz. 8J7 = (1 + 8Z»y (y»^ + ^a^ + a;»y») + (2Z - 5Z*) 17 . iTIT + (-3i» )(irco. e,J7 = 4 (1 + 8Z»)» (y»^ + -?»a^ + «»y») + (-16i + 4/* )U.HU + (-12i« )(^^'. e,,ir= -2(1 + 8P)«(y»^ + -?»a:» + a^y») + (6Z )i;'.irfr + (6Z» )(HU)\ The last preceding table affords a complete solution of the problem, to reduce a ternary cubic to its canonical form. [I add to the present Memoir, in the notation hereof (a, 6, c, /, g^ A, i, j, it, V^x, y, zf for the ternary cubic, some formulae originally contained in the paper "On Homo- geneous Functions of the third order with three variables," (1846), but which on account of the difference of notation were omitted from the reprint, 35, of that paper. Representing the determinant cuc + hy + jz, hx + ki/ + bz, jx + iy + gz, hx + ky + bz, jx +li/ + gz, f kxfby+fz, Ix +Jy + tz, tf Ix -^rfy + iz, gx + %y + cz, f A THIRD MEMOIR UPON QUANTICS. 333 (A, B, C, F, 0, E^x, y, zf alues of -4, B, (7, F, 0, H (equations (10) of 35) are jover writing lat B G H ^ Cf ^ - p • bi - r - i* be -fi fg + ek -2U ki + bg -2fl ag - J' hi - p - f • ■ V + eh -2gl ea - SO gh + M -2jl ok - A» bh - P ¥ + bj -2kl + a/ -2hl ah - hk 2M -2al 2kl -2V" 2gl -2t>- P + gk - hi gh — ai - «/ 2hl -2jk 2/k -2U 2*; -2/& ki - hg P + hi -Jj - gi^ ¥ - y 2jl -2gh 2/7 -2ki 2^ -2el f9 - ek • • - cA p - gk - hi FU^' a, k, g, I, j, h, h, h, i, f, I, k, j, f, e, i, g, I, 2f . . . K V A, B, C, F, G, H FU = A& + Eo + Cc + 2Fi+ 2(?g + afHi, 334 A THIRD MEMOIR UPON QUAKTICS. [14 then the values of a, b, c, f, g, h (equations (13) of 35) are (?e (W eni w ^c 2c/ -2t« 2bi -2/» - 6c 2q; -2g> 2a5r -2/ si — ca 2hh -2P 2aA; -2A» Aii; - ab -6y -2cA 4A; -4a/ 2/ -2a^ 3at ^2jl ca - 93 Afk -46/ Shi -6; A; -2a/ a6 2A:« -26A -2kl - ¥ 4^i -4c/ 8// -26^ 3cA; -2i/ 6c 2»' -2c/ Ski -26; Ahj -4a/ 2A» -2a^ ab - hk 3a/ -2« 8i/ -2cA; 4/A; -46/ 36<7 -2/1 - ki ca 2/« -26* 6e - f* Agi -4c/ Sjl -Qgh -2ai 3cA -2gl • m - « 2p» -2c; 6Ai + 6/i -4(7A; -8/» 2asr -2/ 2aA; -2A* Aal -Ahj + 2AZ -3a/ + 2j/ -3a» 26t -2/» 6^' -4A» 26A ¥ + 2kl -3bj 46/ -4/* Id + 2/1 -Sbg 2c/ -2i^ 2c;- -2<^ 6gk -SP m m V + 2gl -3ch + 2t/ -3ck id -igi 2/i -26c 4cA + 4(7/ -8t; 46; + 4A:/ -8/t/- AP + 2hi -Sgk -6/1 - hg ifg - dc 4db 4-4t/ -2ca 4a/ + 4A/ -8iA; 7gh - 6jl — ai 4P + 2y* -8A» 7v -6gl — ch • 46(7 + 4// -8A:t Aai + Ajl + Sgh 2M -2a6 7jk -ehi - ¥ 7A/ 4P + 2y* + 2Ai -8;5f 144] A THIRD MEMOIR UPON QUANTIC8. 335 Also if the discriminsat be written K(U) = a k 9 I • 3 h h b ■ I f I k • • c • X 9 I ^ m 05 % % n n iS I s % m aj I OD { as % then the values of a, 33, ®, Jp, ffi, |^, I, 2I» ^> 'I (equations (20) of 35) are a = a% + 2hjl " aP - gh^ - j^k, 33 = 6iA + 2/ikf - 6P - A/« - Jfc^i, (!C=(2/' +25n7 - cfi - /^r^ - i^;, 3JF = 6cA + K; - cA:» +2gfk-2bgl+fP ^pj - /lA, 3ffi = ca/ + cjk - ai« + 2A5ri - 2cAi + 5rP -^r'A? - 53/; 3|^ = a65r+ aJfci - 6f +2fhj ''2afl + hl^ -hH - AJfc^r, 3 I = 6c; + c/t - 65r» + 2% - 2ckl -hjl^ - i»A - fij, 33J = CO* + oflr/- cA> + 2ijh - 2aa + kl^ -ff - a/Jfc, 3!a = aW + bhg" ap + 2j*/ - 2hjl^il^ -k^g - hki, 61 = ofcc + 3^A + Sijk +2/* - a/i- bgj-chk - 2Z5fA; - 2iAi - 2ljj, The equation JST ( IT') = i2 = 64iS* — T* would however afford a perhaps easier formula for the calculation of the discriminant.] 336 [145 145. A MEMOIR UPON CAUSTICS. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 273—312. Received May 1,— Read May 8, 1856.] The following memoir contains little or nothing that can be considered new in principle; the object of it is to collect together the principal results relating to caustics in piano, the reflecting or refi::acting curve being a right line or a circle, and to discuss, with more care than appears to have been hitherto bestowed upon the subject, some of the more remarkable cases. The memoir contains in particular researches relating to the caustic by refraction of a circle for parallel rays, the caustic by reflexion of a circle for rays proceeding from a point, and the caustic by refiuction of a circle for rays proceeding from a point; the result in the last case is not worked out, but it is shown how the equation in rectangular coordinates is to be obtained by equating to zero the discriminant of a rational and integral function of the sixth degree. The memoir treats also of the secondary caustic, or orthogonal trajectory of the reflected or refracted rays, in the general case of a reflecting or refracting circle and rays proceeding from a point; the curve in question, or rather a secondary caustic, is, as is well known, the Oval of Descartes or 'Cartesian': the equation is discussed by a method which gives rise to some forms of the curve which appear to have escaped the notice of geometers. By considering the caustic as the evolute of the secondary caustic, it is shown that the caustic, in the general case of a reflecting or refracting circle and rays proceeding from a point, is a curve of the sixth class only. The concluding part of the memoir treats of the curve which, when the incident rays are parallel, must be taken for the secondary caustic in the place of the Cartesian, which, for the particular case in question, passes off to infinity. In the course of the memoir, I reproduce a theorem first given, I believe, by me in the Philosophical Magazine, viz. that there are six different systems of a radiant point 145] A MEMOIR UPON CAUSTICS. 337 and refracting circle which give rise to identically the same caustic, [see post, xxviii]. The memoir is divided into sections, each of which is to a considerable extent in- telligible by itself, and the subject of each section is for the most part explained by the introductory paragraph or paragraphs. I. Consider a ray of light reflected or refracted at a curve, and suppose that ^, 17 are the coordinates of a point Q on the incident ray, a, fi the coordinates of the point Q of incidence upon the reflecting or refracting curve, a, b the coordinates of a point N upon the normal at the point of incidence, x, y the coordinates of a point q on the reflected or refracted ray. Write for shortness, (6-y3)(f-a)-(a-a)(,,-^)= VQQN, (a - a) (f - a) + (6 - /S) (i, - /S) = U^QN, then ^QGN is equal to twice the area of the triangle QON, and if f, 17 instead of being the coordinates of a point Q on the incident ray were current coordinates, the equation VQGN=0 would be the equation of the line through the points G and N, Le. of the normal at the point of incidence ;* and in like manner the equation □QG^i\r = would be the equation of the line through G perpendicular to the line through the points G and -AT, ie. of the tangent at the point of incidence. We have and therefore identically. NG =(a-a)» + (6-)8)«, W 'QG'='^QG^+ dqgn\ Suppose for a moment that <f> is the angle of incidence and <f>' the angle of reflexion or refraction; and let fi be the index of refraction (in the case of reflexion /Lt = — 1)^ then writing (6-y8)(a;-a)-(a-a)(y-;8)=V5rGi\r, and we have 56« = (a:-a)« + (y-i8)^ . ^ VQGN . ^, VqGN ''''''' = NGTGQ' ^^^*=^GTG^' and substituting these values in the equation sin* (fy-fi^ sin* 0' = 0, C. II. 43 338 A MEMOIR UPON CAUSTICS. [l45 we obtain ^* V^GN' - fi' Q^VqGN^ = 0, an equation which is rational of the second order in x, y, the coordinates of a point q on the refracted ray; this equation must therefore contain, as a factor, the equation of the refracted ray; the other factor gives the equation of a line equally inclined to, but on the opposite side of the normal; this line (which of course has no physical existence) may be termed the false refracted ray. The caustic is geometricallt/ the envelope of the pair of rays, and for finding the equation of the caustic it is obviously convenient to take the equation of the two rays conjointly in the form under which such equation has just been found, without attempting to break the equation up into its linear factors. It is however interesting to see how the resolution of the equation may be effected; for this purpose multiply the equation by Nffi, then reducing by means of a previous formula, the equation becomes C^^GN' + a^Qlt)VQGN^ - A'^W^^ + aQGN')'7^N' = 0, which is equivalent to and the factors are Vj(?iV\//A«DQGy + (M— l)VQGi7" TDqGN.VQGN^O; it is in fact easy to see that these equations represent lines passing through the point G and inclined to GN at angles ± <t>\ where 0' is given by the equations sm<f> = fi sin <f>\ and there is no difficulty in distinguishing in any particular case between the refracted ray and the false refracted ray. In the case of reflexion /4* = — I, and the equations become ^qGN. DQGN+ UqGN . VQGN^O; the equation VqGN, DQGN-OqGN. VQGN^O is obviously that of the incident ray, which is what the false refi*acted ray becomes in the case of reflexion ; and the equation VqGN . DQGN-hDqGN. S/QGN = is that of the reflected ray. 145] A MEMOIR UPON CAUSTICS. 339 11. But instead of investigating the nature of the caustic itself, we may begin by finding the secondary caustic or orthogonal trajectory of the refracted rays, i.e. a curve having the caustic for its evolute; suppose that the incident rays are all of them normal to a certain curve, and let Q be a point upon this curve, and considering the ray through the point Q, let G be the point of incidence upon the refracting curve ; then if the point G be made the centre of a circle the radius of which is fjT^ . GQ, the envelope of the circles will be the secondary caustic. It should be noticed that, if the incident rays proceed from a point, the most simple course is to take such point for the point Q. The remark, however, does not apply to the case where the incident rays are parallel ; the point Q must here be considered as the point in which the incident ray is intersected by some line at right angles to the rays, and there is not in general any one line which can be selected in preference to another. But if the refracting curve be a circle, then the line perpendicular to the incident rays may be taken to be a diameter of the circle. To translate the construction into analysis, let f, rj be the coordinates of the point Q, and a, 13 the coordinates of the point G, then f, 17, a, ^8 are in efifect functions of a single arbitrary parameter ; and if we write then the equation where x, y are to be considered as current coordinates, and which involves of course the arbitrary parameter, is the equation of the circle, and the envelope is obtained m the usual manner. This is the well-known theory of Gergonne and Quetelet. III. There is however a simpler construction of the secondary caustic in the case of the reflexion of rays proceeding from a point. Suppose, as before, that Q is the radiant point, and let G be the point of incidence. On the tangent at G to the reflecting curve, let fall a perpendicular from Q, and produce it to an equal distance on the other side of the tangent; then if q be the extremity of the line so produced, it is clear that g is a point on the reflected ray Gq, and it is easy to see that the locus of J is the secondary caustic. Produce now QG to a point Q' such that GQ' = QGy it is clear that the locus of Q' will be a curve similar to and similarly situated with and twice the magnitude of the reflecting curve, and that the two curves have the point Q for a centre of similitude. And the tangent at Q' passes through the point 5, Le. q is the foot of the perpendicular let fall from Q upon the tangent at Q\ we have therefore the theorem due to Dandelin, viz. 43—2 340 A MEMOIR UPON CAUSTICS. [l45 If rays proceeding from a point Q are reflected at a curve, then the secondary caustic is the locus of the feet of the perpendiculars let fall from the point Q upon the tangents of a curve similar to and similarly situated with and twice the magni- tude of the reflecting curve, and such that the two curves have the point Q for a centre of similitude. IV. If rays proceeding from a point Q are reflected at a line, the reflected rays will proceed from a point q situate on the perpendicular let fall from Q, and at an equal distance on the other side of the reflecting line. The point q may be spoken of as the image of Q; it is clear that if Q be considered as a variable point, then the locus of the image q will be a curve equal and similar but oppositely situated to the curve, the locus of Q, and which may be spoken of as the image of such curve. Hence it at once follows, that if the incidental rays are tangent, or normal, or indeed in any other manner related to a curve, then the reflected rays will be tangent, or normal, or related in a corresponding manner to a curve the image of the first- mentioned curve. The theory of the combined reflexions and refractions of a pencil of rays transmitted through a plate or prism, is, by the property in question, rendered very simple. Suppose, for instance, that a pencil of rays is refracted at the first surface of a plate or prism, and after undergoing any number of internal reflexions, finally emerges after a second refraction at the first or second surface; in order to find the caustic enveloped by the rays after the second refraction, it is only necessary to form the successive images of the first caustic corresponding to the different reflexions, and finally to determine the caustic for refraction in the case where the incident rays are the tangents of the caustic which is the last of the series of images; the problem is not in effect different from that of finding the caustic for refraction in the case where the incident rays are the tangents to the caustic after the first re- fraction, but the line at which the second refraction takes place is arbitrarily situate with respect to the caustic. Thus e.g. suppose the incident rays proceed from a point, the caustic after the first refraction is, it will be shown in the sequel, the evolute of a conic; for the complete theory of the combined reflexions and refractions of the pencil by a plate or prism, it is only necessary to find the caustic by refraction, where the incident rays are the normals of a conic, and the refracting line is arbitrarily situate with respect to the conic. V. Suppose that rays proceeding from a point Q are refracted at a line; and take the refracting line for the axis of y, the axis of x passing through the radiant point Q, and take the distance QA for unity. Suppose that the index of refraction /a is put equal to r. Then if ^ be the angle of incidence and <(/ the angle of refraction. 145] A MEMOIR UPON CAUSTICS. 341 we have sin <f>' = k sin <t>, and the equation y — x tan <f/ = tan <t> of the refiracted ray becomes, putting for ^' its value, y — 7-==== X — tan = 0. ^ Vl-Jfc«sin«<^ ^ Differentiating with respect to the variable parameter and combining the two equations, we obtain, after a simple reduction, kx=-' {l-k' sin* <^)* cos*^ A'* sin' *'^""~ cos»0 ' where i' = Vl — A» ; hence eliminating, (Aa:)*-(A'y)*=l, which is the equation of the caustic. When the refraction takes place into a denser medium k is less than 1, and k'^ is positive, the caustic is therefore the evolute of a hyperbola (see fig. 1); but when the refraction takes place in a rarer medium k is greater than 1, and A'* is negative, the caustic is therefore the evolute of an ellipse (see fig. 2). These results appear to have been first obtained by Gergonne. The conic (hyperbola or ellipse) is the secondary caustic, and as such may be obtained as foUowa VI. The equation of the variable circle is aj" 4- (y - tan <^)» - A:" sec" = ; or reducing, the equation is ^ ^y* - 2y tan <^ + A'« tan«<^- ifc« = : whence, considering tan as the variable parameter, the equation of the envelope is A:'«(aj« + y«-*»)-y» = 0, that is, jf'ar' - Jfcy - A^^jfc'" = 0, A HiaiOIR UPON CAUSTICS. is the equation of the secondary caustic, or conic haviog the caustic for its evol The radiant point, it is clear, is a focus of the conic VII. Let the equation of the refracted ray be represented by Xa:+Yy + Z = 0, ve have from which we obtain Vl - ft» sinV 3> F'~2» for the tangential equation of the caustic ; or if we represent the equation of refracted ray by Zx+Yy-k = 0. then we have X' 7' f for the tangential equation of the caustic Fig. 1. Fig. 2. m A HBUOtR UPON CAUSTICS. vin. If a ray be reflected at a circle; we may take a, 6 as the coordinates of the centre of the circle, and supposing as before that f, 17 are the coordinates of a point Q in the incident ray, a, j8 the coordinates of the point of incidence, and x, y the coordinates of a point q in the reflected ray, the equation of the reflected ray, treating 3;, y as current coordinates, is ((i-/3)(«-.)-(o-.)(y-»n(a-.)(f-.) + (i-«(l-S)l + !(«-«)(«-.) +(i-«(y-(J))Ki-»(t-.)-(a-.)(,-^)).0. Write for shortness, *,..-('-»(«-«)-(»-a)(y-«. T,., .(a-i.)(»-a) + (i-S)(y-ffl, and similarly for Nf^^a &c. ; the equation of the reflected ray is Suppose that the reflected ray meets the circle agun in Q' and undergoes a second reflexion, and let jb', y* be the coordinates of a point q' in the ray thus twice reflected. We see first ((?' being a point in the first reflected ray) that Again, considering G as a point in the my by the reflexion of which the second reflected ray arises, the equation of the second reflected ray is uid from the form of the expressions Nq,a, '^q.a ^^ ^ clear that he equation for the second reflected ray may therefore be written under the form r reducing by a previous equation, we obtain finally for the equation of the second iflected ray, td in like manner the equation for the third reflected ray is 1 n on, the equation for the last reflected ray containing, it will be observed, the tdinates of the radiant point and of the first and last points of incidence (the ^dinates of the liist puitit of incidence can of course only be calculated &ora those : radiant point and thu first point of incidence, through the coordinates of the iate pqinU of incidence), but not containing explicitly the coordinates of any intermediate points of incidence. The form ia somewhat remarkable, but the I Is nal^ the same with that obtained by simple geometrical considerations, as ^ 344 A MEMOIR UPON CAUSTICS. [145 IX. Consider a ray reflected any number of times at a circle; and let G^G, be the ray incident at G, and GG' the last reflected ray, the point at which the reflexion takes place or last point of incidence being G. Take the centre of the circle for the origin, and any two lines Ox, Oy through the centre and at right angles to each other for axes, and let Ox meet the circle in the point A, Write /.AOG, =^0, Z.xG,G, = iro, ZAOO =0, ZxGG' =i/r, z GoG,0 = <l> ; then the radius of the circle being taken as the centre of the circle, the equation of the reflected ray is y — sin ^ = tan '^ (a? — cos 0) ; and if there have been n reflexions, then =^o + w(7r-2<^) = ^o +n7r-2n<^, •^ = '^o — 2n0, and therefore the equation of the reflected ray is y cos (iiro - 2n0) - x sin (i/to - 2n<^) + (-)** sin (-^o - ^o) = 0. X. If a pencil of parallel rays is reflected any number of times at a circle, then taking AO for the direction of the incident rays, we may write ^o = 0, '^o = 'n", and the equation of a reflected ray is X sin 2n<l> + y cos 2n^ = (—.)'* sin ^ ; 145] A MEMOIR UPON CAUSTICS. 345 differentiating with respect to the variable parameter, we find X cos 2n^ — y sin 2n^ = (— )** 5- cos ^ ; and these equations give X = ^^ I (2n + 1) cos (2n - 1) <^ - (2n - 1) cos (2w + 1) ^l , y = ^-^ I - (2n + 1) sin (2n - 1) + (2n - 1) sin (2w + 1 ) </>! , which may be taken for the equation of the caustic; the caustic is therefore an epicycloid: this is a well-known result. XL If ra3r8 proceeding from a point upon the circumference are reflected any number of times at a circle, then taking the point A for the radiant point, we have ao = 0, ^^ = 'jr — <f>, and the equation of a reflected ray is a;sin(2n+ l)0-f y cos (2n + l)0 = (—)** sin <f>\ differentiating with respect to the variable parameter, we find ^cos(2r{ + 1)0 — y sin(2n+ 1)0 = ( — )*»- -sin0; and these equations give (-)» ( ) a? = o '^ < (n + I)cos2n0— wco8(2n + 2)0- , (-Y ( ) y-^-Zi j-(w + l)8in2ii0 + ncos(2n + 2)0 which may be taken as the equation of the caustic ; the caustic is therefore in this case also an epicycloid: this is a well-known result. XII. CJonsider a pencil of parallel rays refracted at a circle; take the radius of the circle as unity, and let the incident rays be parallel to the axis of x, then if 0, 0' be the angles of incidence and refraction, and /n or t be the index of refraction, so that sin 0^ = A; sin 0, the coordinates of the point of incidence are cos 0, sin 0, • and the equation of the refracted ray is y — sin = tan (0 — 0') (x — cos 0), le. cos (0 — 0O(y — sin 0) = sin (0 — if}') (x — cos 0), c. II. 44 346 A MEMOIR UPON CAUSTICS. [145 or y cos {<f> — 0') — a? sin (^ — if>) = sin 0', which may also be written (y cos <^ — a? sin 0) cos 0' + (y sin + a? cos — 1) sin ^' = 0. Writing k sin ^, vT— A;* sin' j> instead of sin if/, cos <f>\ and putting for shortness y cos — a? sin = F, y sin + a? cos <^ = X, A? sin _ . Vl - ^> sin* <^ the equation of the refracted ray becomes F+4>(Z-1) = 0; and differentiating with respect to the variable parameter 0, observing that dF_ y dX _ «. d<^ ' d(f> d4>_ icos<<^ _ cot^ . # " (1 - A:> sin«^)* "" 1^ A:* sin* if> ' we have V 1 - if 81U' / ' 4> and the combination of the two equations gives y.^_^(lj-A^8in»^ <Pcot</>-l ' ^ _ 4> c ot (^ — A;* si n' 4>cot<^-l " • and we have therefore TT ^ , v • ^ Ar'sm'^^^coti^-l) , . . , . V=xcos6 + XsinA= ^\ , — r ^ = A:^ sin* 6, ^ ^ ^ 4>cot<^ — 1 ^ * ( -. — 7 - A;* sin* 6 I - A;* sin* 6 cos 6 a? = Xcos6— Fsm A= ^^- -g- ; — ; ^ ^ 4> cot <^ - 1 i.e. _ ^ (1 — A;* sin* <b) — A:* sin* <^ cos ^ ~ <P cos — sin ' 145] A MEMOIR UPON CAUSTICS. 347 or multiplying the numerator and denominator by (1 — A:* sin* <^) (4> cos ^ + sin <^), the numerator becomes (1 - A:* sin* <^) {** cos <^ (1 - A;* sin* <^) - A:* sin* <^ cos if} + * (sin <^ (1 - A;* sin* 0) - A:* sin* ^ cos 0)} = A:* sin* </> cos {(1 - A;* sin* 0) - sin* (1 - A;* sin* 0)} + k sin* Vl - A:* sin* ^ (1 - A;* an* 0) = A* sin* cos* + A? sin* ^ (1 - A:* sin* 0)*, and the denominator becomes A;* sin* ^ cos* <^ — (1 — A;* sin* 0) sin* <f) = - A/* sin* </>, if A/* = l-A:*. Hence we have for the coordinates of the point of the caustic, [A/«a? = - A:* cos* <^ - A; (1 - A?* sin* <^)*, y= A?* sin* ^ ; and eliminating 0, we obtain for the equation of the caustic, A;'«ar = -A^{l-.A;-*y*}*-A;{l-A;V}*; or writing - instead of k, we find (i-/.*)^=(i-/iV)*+/^(i-/*"*y¥ for the equation of the caustic by refraction of the circle, for parallel raya The equation was first obtained by St Laurent. XIII. The discussion of the preceding equation presents considerable interest. In the first place to obtain the rational form write a =(i-/iO^, )8 = (i-/.V)^7=/^(i-/*-*y¥; this gives a* - 20? (/S" + 7*) + (/S* - 7')' = 0, and we have /3« = 1 - 3/i V + 3/tV - yity, rf ^fi*- 3/t%* + 3/t V - y», and consequently /3« - y = (I - /I') {1 - 3/iV + (!+/*') y'i- 44—2 348 A MEMOIR UPON CAUSTICS. [l45 Hence dividing out by the factor (1 — fi^y, the equation becomes (l-/Lt«)«a;*-2(l+;i«-6AtV+3/^*(l+/**)y*-(l+/^')j^)2a?" + (l-3AtV + ^^ or reducing and arranging, + (12/tV-h9AtV)y^-(6/^*(l+/^*)a^ + 6At* + 6At*(l+At»)y»)y* = 0, which is of the form A + 3/Lt*5y* - QfjfiCy* = ; and the rationalized equation is 4* + 27 fi*B'i/' - 21 6/i«(?y* + Bi^fi^ABCy* = 0, where the values of A, B, C may be written 4 = (^ + y»){(l-/.«)»^ + (H-/i»)»y»}-2(l+/.»)(a;»-y») + l, 5 = 4«r« + 3y«, the caustic is therefore a curve of the 12th order. To find where the axis of x meets the curve, we have y=0, 4o' = 0, where = {(!-,.)» a;--l}{(l+/i)»^-l}, i.e. fy-O. ^ X— ± = , ir= + 1 " fi' 1 4-/a' or there are in all four points, each of them a point of triple intersection. To find where the line oo meets the curve, we have where i.e. ^ 1 + /i» . 145] A MEMOIR UPON CAUSTICS. 349 or the curve meets the line oo in four points, each of them a point of triple intersection: two of these points are the circular points at oo. To find where the circle as* + y* = 1 meets the curve, this gives a;* = 1 — y*, and thence 5 = 4-y», and the equation becomes {/A« (/*' - 4) + 4 (1 + 2/i«) y«}« + 27/A* (4 - y'/ y» - 216 (/A« + 2)» y* + 54At»(Ai« + 2)y»(4-y»){/Lt«(Ai«-4) + 4(l+2At»)y»}=0, which is only of the eighth order; it follows that each of the circular points at oo (which have been already shown to be points upon the curve) are quadruple points of intersection of the curve and circle. The equation of the eighth order reduces itself to (y»-/iy{27Aiy + (M'-4)»}=0; the values of x corresponding to the roots y = + /a are obtained without difficulty, and those corresponding to the other roots are at once found by means of the identical equation (/Lt«-4)» + 27/i* + (l-/i»)(^« + 8)» = 0; we thus obtain for the coordinates of the points of intersection of the curve with the circle a;* + y' = l, the values a? = ± Vl - ^» foo, |a?=±V: (a?=±ty, (y=±/f, x= ± = t, ^ (m' - 4)* . each of the points of the first system being a quadruple point of intersection, each of the points of the second system a triple point of intersection, and ectch of the pomts of the third system a single point of intersection. 1 1 Next, to find where the circle a;*4-y*=r— meets the curve; writing a;* = — — y», I* fit we obtain for y an equation of the eighth order, which after all reductions is (y* - ^)* {27/ty + (1 - v/} = 0, 350 A MEMOIR UPON CAUSTICS. [145 and we have for the coordinates of the points of intersection, ( {:= 00 a? = ± iy, X 1 (1 + 8m') «= + y = ^ (1 - V)* • ?= ». each of the points of the first system being a quadruple point of intersection, each of the points of the second system a triple point of intersection, and each of the points of the third systemi a single point of intersection. The points of intersection with the axes of a?, and the points of triple inter- section with the circles aj" + y'=l and a;*-fy« = — , are all of them cuspidal points; the two circular points at oo are, I think, triple points, and the other two points of intersection with the line oo, cuspidal points, but I have not verified this: assuming that it is so,' there will be a reduction 54 accounted for in the class of the curve, but the curve is, in fact, as will be shown in the sequel, of the class 6 ; there is consequently a reduction 72 to be accounted for by other singularities of the curve. XIV. It is obvious from the preceding formulae that the caustic stands to the circle^^ radius -, in a relation similar to that in which it stands to the circle, radius 1, Le. to the refiracting circle. In fact, the very same caustic would have been obtained if^^ the circle radius — had been taken for the refracting circle, the index of refraction being ~ instead of ^. This may be shown very simply by means of the irrational form of the equation as follows. The equation of the caustic by refraction of the circle, radius 1, index of refraction /ir, is as we have seen (i-/.>)fl:=(i-/iW+M(i-/*-»y¥; hence the equation of the caustic by refiuction of the circle radius o\ index of refiwjtion /i', is (.-.^,i.{.-.-.(i)'}'...{i-.-.(|)y, 145] A MEMOIB UPON CAUSTICS. 351 or, what is the same thing, which becomes identical with the equation of the first-mentioned caustic if /a' = c' = — . Hence taking c instead of 1 as the radius of the first circle, we find, Theorem. The caustic by refi-action for parallel ra}rs of a circle, radius c, index of refraction /i, is the same curve as the caustic by refraction for parallel rays of a concentric circle, radius - , index of refraction - . XV. We may consequently in tracing the caustic confine our attention to the case in which the index of refiuction is greater than imity. The circle, radius -, will in this case be within the refracting circle, and it is easy to see that if from the extremity of the diameter of the refiucting circle perpendicular to the direction of the incident rays, tangents are drawn to the circle, radius - , the points of contact are the points of triple intersection of the caustic with the last-mentioned circle, and these point-s of intersection being, as already observed, cusps, the tangents in question are the tangents to the caustic at these cusps. The points of intersection with the axis of X are also cusps of the caustic, the tangents at these cusps coinciding with the axis of x: two of the last-mentioned cusps, viz. those whose distances from the centre are 1 . . . . . c ± , lie within the circle, radius - , the other two of the same four cusps, viz. M + 1 M those whose distances from the centre are ± M-l , lie without the circle, radius - ; the last-mentioned two cusps lie without the refracting circle, when /a < 2, upon this circle, when /a = 2, and within it and therefore between the two circles, when /A>2. The caustic is therefore of the forms in the annexed figures 3, 4, 5, in each Fig. 3. Fig. 4. Fig. 5. T"-^--. **.. 352 A MEMOm UPON CAUSTICS. [145 of which the outer circle is the refracting circle, and fi ib > 1, but the three figures correspond respectively to the cases ft < 2, ft = 2 and /k > 2. The same three figures will represent the different forms of the caustic when the inner circle is the refracting- circle and fi is < 1, the three figures then respectively corresponding to the cases A^ > i> /^ = i> ^^ /* < i- XVI. To find the tangential equation, I retain k instead of its value -; the equation of the refructed ray then is a? (A; cos - Vl - ^in« <^) + y (k sin <^ + cot ^ Vl - A:» sin* <t>) - Jfc = 0, and representing this by Zar+Fy-A;=0, we have Z = A;cos0-Vl-A;»sin»<^, F = A: sin ^ 4- cot ^ Vl — A:* sin* 0, equations which give X cos + Fsin ^ = A?, z«+r* ^ and consequently sin* <l> ' . . 1 sm = ,— , Vz*+y*-i cos 6 = — .— — - ^ VZ* + F* ^ ) and we have which gives ZVZ*+ F*-l + F-ifc VZ*+ F* = 0, (Z«+F*)(Z*-l-Jfc*) = -2A;FVZ*+F*; or, dividing out by the factor VZ* + F*, the equation becomes \/Z*+F*(Z*- 1-Jfc*) = - 2A;F, from which (Z*+F*)(Z*-l-Jfc*)*-4ifc»F* = 0; or reducing and arranging, we obtain Z*(Z*-l-Aj*)*+F*(Z + l+A;)(Z + l-A:)(Z-l+ifc)(Z-l-i) = for the tangential equation of the caustic by refraction of a circle for parallel raj^s. The caustic is therefore of the class 6. 145] A MEMOIR UPON CAUSTICS. 353 XVII. Suppose next that rays proceeding from a point are reflected at a circle. A very elegant solution of the problem is given by Lagrange in the Mim. de Turin; the investigation, as given by Mr P. Smith in a note in the Cambridge and Dublin Mathematical Journal, t. ii [1847] p. 237, is as follows: Let B be the radiant point, RBP an incident ray, and PS a reflected ray; CA a fixed radius; ACP = a, ACB = €, reciprocal of CB = Cy reciprocal of CP = a, The equations of the incident and reflected ray, where w = - , may be written u=^Asin0 + B cos ; incident ray, t* = i4sin(2a — ^) + 5cos(2a — ^); reflected ray, the conditions for determining A and B being a = ^ sin a 4- 5 cos a, c = -A sin e + -B cos e, whence . _ a cos € - c cos a p _ c sin a — a si n e sm (a — e) sin (a — e) Substituting these values, the equation of the reflected ray becomes a sin (2a — ^ — e) = M sin (a — 6) 4- c sin (a — ^), firom which and its differential with respect to the arbitrary parameter ce, the equation of the caustic, or envelope of the reflected rays, will be found by eliminating a. In this, a being the only quantity treated as variable in the differentiation, let 2a-^-e = 20, therefore a iL 45 354 A MEMOIR UPON CAUSTICS. [l45 and the equation becomes a sin 20 = w sin {<^ + J (^ - €)} + c sin [<f>-^{0- e)}. Make also p __ (<f + c) cos ii^ — e) ^ 2a ] 1 then the equation becomes with the condition Hence cos ' ^ sin (^ ' P^ + Qy = i. •»"'+y"*=i. multiplying by x and y, and adding, we find X = 1 ; therefore Hence or restoring the values of P and Q, {(16 + c) cos i (^ - e)}* + {(u - c) sin ^{0- e)}* = 1, the equation of the caustic. XVIII. But the equation of the caustic for rays proceeding from a point and reflected at a circle may be obtained by a different method, as follows: Take the centre of the circle for origin ; let c be the radius of the circle, fl, ^ the coordinates of the radiant point, a, the coordinates of the point of incidence, X, y the coordinates of a point in the reflected ray. Then we have from the equation of the circle o^-\-^'=^&, and the equation of the reflected ray is by the general formula, (ft'x - a/8) (cm? + /8y - c») + (ya - a?/8) (aa + 6/8 - c*) = ; 145] A MEMOIR UPON CAUSTICS. 355 or arraDging the terms in a diflferent order, (6a? + ay)(a»-)8') + 2(6y-cur)a^-c^(6+y)a + c»(a + a?))8 = 0; and writing herein a = c cos 0, /8 = c sin ^, the equation becomes (bx + ay) cos 2^ + {hy — olx) sin 2^ — (6 + y) c cos ^ + (a + a:) c sin ^ = 0, where d is a variable parameter. Now in general to find the envelope of -A cos 2^ + J5 sin 2^+ C cos ^ + 2) sin ^ + ^ = 0, we may put e^==z, which gives the equation and equate the discriminant to zero : this gives (4/)» - 27 (- 8J)» = 0, where -8J = 4((?-2>»)+25C'2)--{8(^'' + £«) + ((? + 2>»)}i^+^^, and consequently - 27 [A ((?- Z)«) + 2BGD - (8 {A^^B") + (0+ Z>«)) ^^ + ^J5?}« = 0; «nd substituting for Ay B, C, D, E their values, we find f 4 (a« + 6») (aj" + y*) - c« ((a? + a)« + (y + 6)»)}'- 27 (bx-ayy (a^ + f- a^^¥y=- 0, ibr the equation of the caustic in the case of rays proceeding from a point and xeflected at a circle: the equation was first obtained by St Laurent. It will be convenient to consider the axis of ^ as passing through the radiant point ; this gives 6 = 0; and if we assume also c = l, the equation of the caustic becomes {(4a« - 1) (aj" + y«) - 2gw? - a«}» - 27ay (ic» + y»- a»)» = 0. XIX. Reverting to the equation of the reflected ray, and putting, as before, c = 1, 6 = 0, tihis becomes / « /I . 1 \ a cos 2^ — cos ^ ^ (-2acos^ + l)a?+ ^— ^ t/ + a = 0; ^ ^ sm r "^ > differentiating with respect to 0, we have (-2asm^)ar + . ,^ y = 0; 45—2 356 and from these equations A MEMOIR UPON CAUSTICS. [■ _ tt^cos (1 -h28 i n«g)~a ^ " 1 -~3a cos 2dT2a« y = 2a« sin» ^ 1- 3a cos 2^+ 2a*' which give the coordinates of a point of the caustic in terms of the angle wl determines the position of the point of incidence. The values in question satisfy, they should do, the equation {(4a» - 1) (aj" + y«) - 2aa? - a»}» - 27ay (ic» + y» - a«)« = 0; we have, in fact, 4a* (cos 0—af ^+y' -tt» = (4a«- 1) (ar» + y«) - 2aa; - a' = (1 - 3a cos 2^ + 2a«)» ' 12aHcos^-ay (l-3acos2^+2a«)>' from which it is easy to derive the equation in question. XX. If we represent the equation of the reflected ray by Xx-^Yy + a- 0, then we have X = -2acos^ + l, a cos 20 — cos F= sin and thence (Z - 1)> - 4a« = - 4a« sin« ^, X'+Y* = -.-\ >, (1 - 2a cos + a»), sm^ 0^ X + a^ =l-2aco80 + a\ and consequently (X*+Y') {(Z- 1)»- 4a»l + 4a»X + 4a* = 0, or, what is the same thing, {Z(Z-l)-2a«}«+F='{(Z-l)«-4a'} = 0, which may be considered as the tangential equation of the caustic by refle: circle; or if we consider X, F as the coordinates of a point, then the equ; be considered as that of the polar of the caustic. The polar is therefore a cv fourth order, having two double points defined by the equations X(Z — 1) — 2a* 145] A MEMOIR UPON CAUSTICS. 357 and a third double point at infinity on the axis of F, i.e. three double points in all ; the number of cusps is therefore 0, and there are consequently 4 double tangents and 6 inflections, and the curve is of the class 6. And as F is given as an explicit function of X, there is of course no difficulty in tracing the curve. We thus see that the caustic by reflexion of a circle is a curve of the order 6, and has 4 double points and 6 cusps (the circular points at infinity are each of them a cusp, so that the number of cusps at a finite distance is 4) : this coincides with the conclusions which will be presently obtained by considering the equation of the caustic. XXI. The equation of the caustic by reflexion of a circle ia {(4a« - 1) (a:» + y«) - 2cw? - a»}» - 27ay (ar» + y* - a^y = 0. Suppose first that y = 0, we have {(4a« - 1) a:* - 2cw? - a*}« = 0, — a __ a ^^' ^'isr+i' ^"2^-r or the curve meets the axis of a? in two points, each of which is a triple point of intersection. Write next aJ" + y'=a^ this gives {(4a« - 1) a« - 2aa? - a«}» = 0, and consequently a? = - a (1 - 2a«), y = ± 2a» Vi - a«, or the curve meets the circle a;* + y*— a^ = in two points, each of which is a triple point of intersection. To find the nature of the infinite branches, we may write, retaining only the terms of the degrees six and five, (4a«- l)'(a;» + y»)»- 6 (4a« - l)«a (a;» + y*)» a?- 27 ay (a;» + y*)^ = ; and rejecting the factor (oc^ + jf^y, this gives (4a»-l)»«»+{(4a«-l)»-27a«}y»-6(4a»-l)»aa: = 0; or reducing, (4a«-l)»a;»-(l-a')(8a» + l)»y»-6(4a*-l)»aa?=0; and it follows that there are two asymptotes, the equations of which are _(4^-_l)*_ L 3^ 1 VT^*(8a' + l)l 4a«-ir 358 A MEMOIR UPON CAUSTICS. [l45 Represent for a moment the equation of one of the asymptotes by y = A{x — o), then the perpendicular from the origin or centre of the reflecting circle is iia-s- Vl -\-A^, and 3a V4a* - 1 Aa = Vl-a«(l+8aO I , ^,^ (l-a')(l+Sa«)' + (4a«-l) »^ 27a' Vl ^A^ = (1 - a«) (1 + 8a»)« (1 - a«) (1 + 8a»)^ ' 3V3a Vl - a« (1 + 8a«) and the perpendicular is ^V4r^»_i, which is less than a if only a«<l, i.e. in every case in which the asymptote is real. The tangents parallel and perpendicular to the axis of x are most readily obtained from the equation of the reflected ray, viz. / « /» 1 \ a cos 2^ — cos ^ ^ {— 2a cos ^+1) a; H : — -^ v + a = 0: the coefficient of x (if the equation is first multiplied by sin^) vanishes if sin = 0, 1 V4a' — 1 which gives the axis of a?, or if cos = — , which gives y = ± — ^ , for the tangents parallel to the axis of x. The coefficient of y vanishes if a cos 20 — cos 5 = 0; this gives ^ 1 + V8a« + 1 . ^ 1 ,- , , ^ /s-5 — ?x cosg= - ^ , 8ine=^- (4a»-l T v8a«+l), 4a oar and the tangents perpendicular to the axis of x are thus given by -2a 1 + V8a« + 1 ' these tangents are in fact double tangents of the caustia In order that the point of contact may be real, it is necessary that sin 0, cos should be real ; this will be the case for both values of the ambiguous sign if o > or = 1, but only for the upper value if a < 1. It has just been shown that for the tangents parallel to the axis of x, we have V4a«-1 y = ± 2a V4a' — 1 the values of y being real for a > ^ : it may be noticed that the value y = — 145] A MEMOIR UPON CAUSTICS. 359 is greater, equal, or less than or to y = 2a' Vl — a', according as a > = or < 7= 5 ^^^ depends on the identity (4a» - 1) - 16a« (1 - a«) = (2a» - 1)« (2a» +1). To find the points of intersection with the reflecting circle, ic* + y' — 1 = 0, we have (3a« - 1 - 2cw:)»- 27a» (1 - a;»)(l - a«>» = ; or, reducing, 8aV + (- 27a* + 18a»- 15) a'ic* + (54a* - 36a» + 6) oo? + (- 27a* + 18a«+ 1) = 0, Le. (aa?-l)»(8aa?-27a*+18a» + l) = 0. The factor (ax^iy equated to zero shows that the caustic touches the circle in the points a? = - , y = ± a/ 1 ^ , i e. in the points in which the circle is met by the polar of the radiant point, and which are real or ima^nary according as a > or < 1. The other &ctor gives 27a* - 18a« - 1 x = 8a Putting this value equal to ± 1, the resulting equation is (a + l)(27a' + 9a + l) = 0, and it follows that x will be in absolute magnitude greater or less than 1, Le. the points in question will be imaginary or real, according as a>l or a<l. It is easy to see that the curve passes through the circular points at infinity, and that these points are cusps on the curve ; the two points of intersection with the axis of X are cusps (the axis of x being the tangent), and the two points of inter- section with the circle a^ + y' — a' = are also cusps, the tangent at each of the cusps coinciding with the tangent of the circle ; there are consequently in all 6 cusps. XXII. To investigate the position of the double points we may proceed as follows: write for shortness P = (4a*-l)(a:* + y')- 2aa?-a*, Q = ar/S, S^-x-k-f-a}) the equation of the caustic is P»-27Q» = 0; hence, at a double point, one of which equations may be replaced by dP dQ^dP dQ^^ dx dy dy dx ' 360 A MEMOIR UPON CAUSTICS. [145 Now — ^£ = 2{(^'-l)x-a], ^| = 2(4a'-l)y. ^=2aay. ^ = o(^ + 3y'-a») = a(S+2y«); substituting these values in the last preceding equation, we find ~(4a»-l)"y "S-h2y«* or, reducing, and using this to simplify the equation we have ^s-'««S-». P'^-lSay8.2axy = 0, i.e. ^ -^ — 9cuF/S> = 0, and therefore Multiplying by P and writing for P* its value 27a^y'S^, we have Px = 3ay«, and thence whence «y* = 5» ^ = S or 2^=?!- a!» • J • S „i' or "a' and substituting in the equation * 4a'-U Sj' we find or, rationalising, 4(M!» - {(4o' - 1) « - o)' = 0, 145] A MEMOIR UPON CAUSTICS. 361 or, what is the same thing, (4flw?-l)(a?-a(a + iVl -a«)») (a:-a(a-iVl -a»)«) = 0. The factor 4tax — l equated to zero gives ^=7- from which y may be found, but the resulting point is not a double point; the other factors give each of them double points, and if we write we find ^ 2aH'(a + iVl-a')* values which, in fact, belong to one of the four double points. It is easy to see that the points in question are always imaginary. It may be noticed, by way of verification, that the preceding values of x, y give (4a« - l)(«" + y') - 2flw: - a» = ^^^^^ (1 - 4a»- 4ai VTIT^), x' + y»-a' = j-j-g^, (3a + i Vl - a« ), f = ^t."^, (- 1 + 14a» - 16a« + 2a (3 - 8a») i Vl - a*) ; and if the quantities within ( ) on the right-hand side are represented by A, B, C, then i (a + iVrr^'). whence we have identically, B f) _ = _(a4-zVl-a»)>, (^)' = 5' - ^'=^^' by means of which it appears that the values of x, y satisfy, as they should do, the equation of the caustic ; and by forming the expressions for (4a'— l)a; — a and a^ + S^*— a', it might be shown, d posteriori^ that the point in question was a double point. XXIII. The equation {(4a«- 1) (a;* + y«) - 2aa? - a«}» - 27ay (a^ + y' - a«)» = l)ecome8 when a = 1 (L e. when the radiant point is in the circumference), {3y> + (a? - 1) (3a?+ 1)}» - 27y« (y» + «» - 1)» = ; it is easy to see that this divides by (a?— 1)*; and throwing out this factor, we have for the caustic the equation of the fourth order, 27 j^ + 18y« (3ar» _ 1) + (a? - 1) (3a; + 1)» = 0. A /• 362 A MEMOIR UPON CAUSTICS. [145 XXIV. The equation {(4a« - 1) (a^ + jf") - 200? - a»}» - 27ay (ic* + y» - a«)» = becomes when a = x (i. e. in the case of parallel rays), (4ic» + 4y«-l)»-27y» = 0. which may also be written 64>af + 48a?* (4y« - 1) + 12a?»(4y» - 1)» + (8y« + 1)» (y* - 1) = 0. XXV. It is now easy to trace the curve. Beginning with the case a = oo , the curve lies wholly within the reflecting circle, which it touches at two points; the line joining the points of contact, being in fact the axis of y, divides the curve into two equal portions ; the curve has in the present, as in every other case (except one limiting case), two cusps on the axis of x (see fig. 6). Next, if a be positive and > 1, the general form of the curve is the same as before, only the line joining the points of contact with the reflecting circle divides the curve into unequal portions, that in the Fig. 7. a>l. Fig. 6. a — CD. ,• i^S^^ 1 ^^^."^^ yy / / j \\ / V [ J \ i f . .-..^....f. \ 1 1 \ ' \ I I J / \ \ • / / *» V 1 y ^ ''C^ Sw neighbourhood of the radiant point being the smaller of the two portions (see fig. 7). When a = 1, the two points of contact with the reflecting circle unite together at the radiant point ; the curve throws off, as it were, the two coincident lines a? = 1, and the order is reduced fi-om 6 to 4. The curve has the form fig. 8, with only a single cusp on the axis of x. If a be further diminished, a < 1 > -= , the curve takes the form v2 shown by fig. 9, with two infinite branches, one of them having simply a cusp on the axis of x, the other having a cusp on the axis of m^ and a pair of cusps at its intersection with the circle through the radiant point; there are two asymptotes equally inclined to the axis of x. In the case a — -j=^^ the form of the curve is nearly the same as before, only the cusps upon the circle through the radiant point lie on the axis of y (see fig. 10). The case a<-~>\ is shown, fig, 11. For = ^, the two 145] A MEMOIR UPON CAUSTICS. 363 asymptotes coincide with the axis of a?; one of the branches of the curve has wholly disappeared, and the form of the other is modified by the coincidence of the asymptotes Fig. 8. a=l. Fig. 13. a = l \ \ \ Fig. 11, ^ 1 -v A- .».,—>.. i X-4- Fig. 10. a V2' Fig. 12. a = J. \ •- — i — with the axis oi x\ it has in fact acquired a cusp at infinity on the axis of x (see fig. 12). When a<\, the curve consists of a single finite branch, with two cusps on the axis of x^ and two cusps at the points of intersection with the circle through the radiant point; one of the last-mentioned cusps will be outside the reflecting circle as long as o>J; fig. 13 represents the case a=^, for which this cusp is upon the reflecting circle. For a<J, the curve lies wholly within the reflecting circle, one of the cusps upon the axis of x being always within, and the other always without the circle through the radiant point, aod as a approaches the curve becomes smaller and smaller, and ultimately disappears in a point. The case a negative is obviously included in the preceding one. Several of the preceding results relating to the caustic by reflexion of a circle were obtained, and the curve is traced in a memoir by the Rev. Hamnet Holditch, (^Tierly MathemcOical Journal, t i. [1857, pp. 93 — 111]. 46—2 364 A MEMOIR UPON CAUSTICS. [145 XXVI. Suppose next that rays proceeding from a point are refraxiied at a circle. Talie the centre of the circle as origin, let the radius be c, and take ^, i; as the coordi- nates of the radiant point, a, /9 the coordinates of the point of incidence, x, y the coordinates of a point in the refracted ray : then the general equation -qG VQON +n*QO '7qON =0 becomes, taking the centre of the circle as the point N on the normal, or writinj^ a=b, 6 = 0, or putting a*-(-y3' = C, and expanding, a' {2 (i,»ar - /iVf )} + a«/8 {- 4 (^x - /i'an/f ) + 2(ryy- ^*fn)\ + a/S* {- 4 (fi,y - ii}xyf,) + 2 (pa: - /iVf)} -o« {(a!» + y» + c»)i?» -M* (?+'?* + C)!^} + 2a^8 ((a^ + y» + c») fv - /*« (f» H- 1;» + c*) ay} -/3» {(«»-l-y' + c»)p-/t«(f» + i7» + c»)a?} = 0, which may be represented by 4a« + P««/8-|- Ca/g' + D/S'-l- .P«» + ^0/9 + H0' = O. Now a* + /8* = c*, and we may write The equation thus becomes -l<-J)'-!«(-l)('-i)-H'-i)'-» or expanding, + -(F-Gi-H) ^ + (3^-£t + C + 3i)i) « + *(F+H) ^=0. + (3il + £» + C - 3IH") - -l--(.P-(-(?i-5) c 1 + (4 + 5i - C + IH) i 145] A MEMOIR UPON CAUSTICS. 365 n which z may be considered as the variable parameter ; hence the equation of the austic may be obtained by equating to zero the discriminant of the above function >f z\ but the discriminant of a sextic function has not yet been calculated. The equation would be of the order 20, and it appears from the result previously obtained or parallel rays, that the equation must be of the order 12 at the least ; it is, I think, )robable that there is not any reduction of order in the general case. It is however )racticable, as will presently be seen, to obtain the tangential equation of the caustic )y refraction, and the curve is thus shown to be only of the class 6. XXVII. Suppose that rays proceeding from a point are refracted at a circle, and let it be •equired to find the equation of the secondary caustic: take the centre of the circle as >rigin, let c be the radius, f , i; the coordinates of the radiant point, a, fi the coordinates )f a point upon the circle, /la the index of refraction; the secondary caustic will be ihe envelope of the circle, where a, /8 are variable parameters connected by the equation a' + zS' — c* = 0; the liquation of the circle may be written in the form But in general the envelope of -4a + £/8 + C = 0, where a, /8 are connected by the equation a' + ^S* — c'ssO, is & {A} + JP) — (? = 0, and hence in the present case the equation of the envelope is which may also be written {/it« (a^ + y» - c*) - (p + 17» - c«)}« = 4c«/it« {(a; - f )« + (y - 17)»}. If the axis of x be taken through the radiant point, then 97 =0, and writing also { = a, the equation becomes or taking the square root of each side, {/A« (a;* + y» - c») - a« + c"} = 2cm V(a?-a)» + y« ; 1 . / IV whence multiplying by 1 — 5 and adding on each side c* ( /^ — ) + (^ — a)^ + y', we have /**|(«-^,y + yj = |V(^-a)« + y« + c(M-J)}', or '^V (^~5) +y'='^(^-«)"+y'+^(^-^)' which shows that the secondary caustic is the Oval of Descartes, or as it will be con- venient to call it, the Cartesian. 366 A MEMOIB UPON CAUSTICS. [145 It is proper to remark, that the Cartesian consists in general of two ovals, one of which is the orthogonal trajectory of the refracted rays, the other the orthogonal trajectory of the false refracted raya In the case of reflexion, the secondary caustic is a Cartesian having a double point ; this may be either a conjugate point, or a real double point arising from the union and intersection of the two ovals ; the same secondary caustic may arise also from refraction, as will be presently shown. XXVIII. Reverting to the original form of the equation of the secondary caustic, multipljring by — J (1 i] and adding on each side ~fl ij +~, {(^ — a)* + y'j, the equation becomes or extracting the square root, Combining this with the former result, we see that the equation may be expressed indifferently in any one of the four forms, ^/(-f)'-^'=^;:^/(^-J)>^'-^- (" - ^) \/(' - 9" - '• ^ (- <■ + ^ \/(' - J.)' - ^ - e - t) ^'<^^^>^ = «■ It follows, that if we write successively a'= a, c' = c f /*' = /* (1) a = - , c = - , M = - (a) a fi a ^ ^ a = - 2 , c = , /A = - (p) a =a , c = - , M = - (7) /x c a=— , C =C y a = — (0) 145] A MEMOIR UPON CAUSTICS. 367 or what is the same thing, (1) (a) (0) (7) or what is again the same thing, a==a' , c = c', t*=l*' a' a' c= -, A* a' of c' 1 a =a' , a' a' a =->, a c = c' , ey a a c' c' a J thing, o' = a. a' a a c'» a a' /*»' a' a a a' ;.'. = « a' = a , c'' a a a a' a a /» a (8) («). (1) («) (/8) (7) (S) (0. we have in each case identically the same secondary caustic, and therefore also identically the same caustic ; in other words, the same caustic is produced by six different systems of a radiant point and refracting circle. It is proper to remark that if we represent the six systems of equations by (a', c', fi!) = (a, c, /i), {a\ c', /i') = a (a, c, /a), &c., then, a, )3, 7, S, 6 will be functional symbols satisfying the conditions a = /S" = &y = eS =76, /9 = a* = 78 = Sc = €7, <y = Sa = a€ = 6/8 = ^S, 5 = ea = 07 = 7/3 =/8€, 6 =:7a = aS=: S/8 =)87. 368 A MEMOIR UPON CAUSTICS. [145 XXIX. The preceding formulae, which were first given by me in the Philosophical Magdzine, December 1853, [124] include as particular cases a preceding theorem with respect to the caustic by refraction of parallel rays, and also two theorems of St Laurent, Oergonne, t. xviii., [1827, pp. 1 — 19] viz. if we suppose first that a=^Cy Le. that the radiant point is in the circumference of the refracting circle, then the system (a) shows that the same caustic would be obtained by writing c, - , 1 (or what is the same thing — 1) in the place of c, c, /x, and we have Theorem. The caustic by refraction for a circle when the radiant point is in the circumference is also the caustic by reflexion for the same radiant point, and for a reflecting circle concentric with the refracting circle, but having its radius equal to the quotient of the radius of the refracting circle by the index of refraction. Next, if we write a = CfjLj then the refracted rays all of them pass through a point which is a double point of the secondary caustic, the entire curve being in this case the orthogonal trajectory, not of the refracted rays, but of the false refracted rays; the formula (S) shows that the same caustic is obtained by writing -, c, 1 (or what is Cv the same thing — 1) in the place of a, c, /i f = - j , and we have Theorem. The caustic by refraction for a circle when the distance of the radiant point from the centre is to the radius of the circle in the ratio of the index of refraction to unity, is also the caustic by reflexion for the same circle considered as a reflecting circle, and for a radiant point the image of the former radiant point XXX. The curve is most easily traced by means of the preceding construction ; thus if we take the radiant point outside the refracting circle, and consider fi as varying fix)m a small to a large value (positive or negative values of /i give the same curve), we see that when fi is small the curve consists of two ovals, one of them within and the other without the refracting circle (see fig. 14). As /i increases the exterior oval continually increases, but undergoes modifications in its form; the interior oval in the first instance diminishes until we arrive at a curve, in which the interior oval is reduced to a conjugate point (see fig. 15); then as fi continues to increase the interior oval reappears (see fig. 16), and at last connects itself with the exterior oval, so as to form a curve with a double point (see fig. 17); and as /i increases still further the 145] A MEMOIR UPON CAUSTICS. 369 curve again breaks up into an exterior and an interior oval (see fig. 18) ; and thence- forward as fjL goes on increasing consists always of two ovals; the shape of the exterior oval is best perceived fh)m the figures. An examination of the figures will also show how the same curves may originate from a different refracting circle and radiant point. Fig. 14. Fig. 17. Fig. 15. Fig. 18. 0. II. 47 370 A MEMOIB UPON CAUSTICS. [l45 XXXI. The theorem, " If a variable circle have its centre upon a circle S, and its radius proportional to the tangential distance of the centre from a circle (7, the envelope is a Cartesian," is at once deducible from the theorem — '' If a variable circle have its centre upon a circle S and its radius proportional to the distance of the centre from a point C\ the locus is a Cartesian,'* which last theorem was in eflfect given in discussing the theory of the secondary caustic. In fact, the hx^us of a point P such that its tangential distances from the circles C, C" are in a constant ratio, is a circle S, Conversely, if there be a circle C, and the locus of P be a circle S, then the circle C" may be found such that the tangential distances of P from the two circles are in a constant ratio, and the circle C may be taken to be a point, i.e. if there be a circle C and the locus of P be a circle S, then a point C may be found such that the tangential distance of F from the circle C is in a constant ratio to the distance from the point C\ Hence treating P as the centre of the variable circle, it is clear that the variable circle is determined in ihe two cases by equivalent constructions, and the envelope is therefore the same in both cases. XXXII. The equatiiJii of the secondary caustic developed and reduced is ^* (x'' + yy - 2/x* (a*-* -\-{fi^+l) c^) (a^ + y^) + 8cV(W 4- a* - 2aV (/x^ + 1 ) + (/x^ - 1 )- c* = ( or, what is the same thing, which niav also be written which is of the form {x" 4- 3/' - ay -h 16.4 (./■ - m) = : and the values of the coefficients are m ^,^ + 0+^)a\ 145] A MEMOIR UPON CAUSTICS. 371 The equation just obtained should, I think, be taken as the standard form of the equation of the Cartesian, and the form of the equation shows that the Cartesian may be defined as the locus of a point, such that the fourth power of its tangential distance from a given circle is in a constant ratio to its distance from a given line. XXXIII. The Cartesian is a curve of the fourth order, symmetrical about a certain line which it intersects in four arbitrary points, and these points determine the curve. Taking the line in question (which may be called the axis) as the axis of x^ and a line at right angles to it as the axis of y, let a, 6, c, d be the values of x corre- sponding to the points of intersection with the axis, then the equation of the curve is # + (.T - a) (x -b)(x- c) (x-d) = 0. It is easy to see that the form of the equation is not altered by writing x-\-0 for x, and a-\-0y b-\-0, c + O, d-\-0 for a, 6, c, d, we may therefore without loss of generality put a-\-b -\- c + d = 0, and the equation of the curve then becomes 3/* + y2 (2^ + aft + ac + ad -h 6c -h M + cd) + (a: - a) (a? - 6) (x - c) (a; - d) = 0, where a-\-b-\-c-\-d = 0', the cur\'e is in this case said to be referred to the centre as origin. The last-mentioned equation may be written (^ + y^y -\-(ab-\-ac + ad-\-bc-\-bd-{-cd){af^ + j/^)- (abc -h aid + acd -f- bed) x + abed = 0, or [a^ -\- y"" -{-^ {db +ac + ad-\-bc -\-bd -\-cd)]' — (abc + ahd + acd -h bed) x -ii = 0, or ( a^¥ + aV + a^'d^ + b^(^ + h'd? + c^d? + 2a26c + 2a26d -h 2a^cd + 26«ac + Wad -h 2b^ed + 2c«a6 + 2c2ad -h 'Id'bd + id'ah -h ^d'ac -h 2d'6c ^ +2a6cd observing that a^be + a'hd H- a^'ed H- b^ae H- 6 W -h Ifed + c^'ab + e^ad + e^'bd 4- d^ab + d^ac -h d?he = a6c (a + 6 + c) H- abd {a-\-b-\-d)-\- acd {a -\- e -{• d) -\-bed{b ^- c ■\- d) 372 A MEMOIR UPON CAUSTICS. [l45 the equation becomes j^i-" + y' + i (a6 + ac + od + 6c + M + cd)}« — (abc + abd + acd + bed) x - \ (a«6« + a^-\-d^d^ + 6«c« + 6«d* + c»d*- 6a6cd) = 0, which is of the fonn (j?* + y« - a)» + 16-4 (a? - 7^1) = 0, and, as already remarked, signifies that the fourth power of the tangential distance a point in the curve from a given circle, is proportional to the distance of the sai point from a given line. The circle in (juestion (which may be called the dirige^^^ circle) has for its equation ^ + y' + i («^ + «c + ad + 6c + 6d + cd) = ; the line in question, which may be called the directrix, has for its equation d}V + a^c^ + a«d« + 6'c^ + h^d? + Cd* - 6a6cd ^ /p j_ — . f I « 4 (oic + abd + ojcd + hcd) ' the multiplier of the distance from the directrix is abc + abd + acd + hcd. It may be remarked that a, 6, c, d being real, the dirigent circle is real; the equation may, in fact, be written ^ + y' = 4 [(a + &)' + (a + c)« + (a + d)« + (6 + c)» + (6 + d)» + (c + d)>]. XXXIV. Considering the equation of the Cartesian under the form (a?> + y' - (jLf + 164 (a: - m) = 0, the centre of the dirigent circle a;* + y' — a = must be considered as a real point, but a may be positive or negative, i.e. the radius may be either a real or a pure imaginary distance: the coefficients A, m must be real, the directrix is therefore a real line. The equation shows that for all points of the curve a? — m is always negative or always positive, according as il is positive or negative, i.e. that the curve lies wholly on one side of the directrix, viz. on the same side with the centre of the dirigent circle if 4 is positive, but on the contrary side if A is negative. In the former case the curve may be said to be an * inside * curve, in the latter an ' outside ' curve. If 7/1 = 0, or the directrix passes through the centre of the dirigent circle, then the distinction between an inside curve and an outside curve no longer exists. It is clear that the curve touches the directrix in the points of intersection of this line and the dirigent circle, and that the points in question are the only points of intersection of the curve with the directrix or the dirigent circle; hence if the directrix and dirigent circle do not intersect, the curve does not meet either the directrix or the dirigent circle. 145] A MEMOIR UPON CAUSTICS. 373 XXXV. To discuss the equation I write first y = 0, which gives for the points of intersection with the axis of x. If this equation has equal roots, there will be a double point on the axis of x, and it is important to find the condition that this may be the case. The equation may be written in the form (3, 0, -a, 12A, 3a»-48ilm$a?, 1)* = 0, the condition for a part of equal roots is then at once seen to be - (a« - l2Amy + (a» - 18il77ia + 54^2)' = ; or reducing and throwing out the factor ^', this is 27 A^ + 2m (8m« -da) A- a« (m« - a) = 0. This equation will give two equal values for A if m» (8m' - 9ay + 27a' (m» - a) = 0, an equation which reduces itself to (4m' - Say = 0. 4m' Hence, if 4wi' — 3a be negative, i.e. if a>-^, the values of A will be imaginary, »j 4m' . 4m' but if 4m' — 3a be positive, or a < - .- , the values of A will be real. If a = —^ , then there will be two equal values of A, which in fact corresponds to a cusp upon the axis of x. Whenever the curve is real there will be at least two real points on the axis of x; and when a<-^, but not otherwise, then for properly selected values of A there will be four real points on the axis of x. Differentiating the equation of the curve, we have {(af^ + f-a)x + 4iA) dx + {a^-tif'- a) ydy = ; and if in this equation we put dx = 0, we find y=0, or ar'-fy' — a = 0, i.e. that the points on the axis of x, and the points of intersection with the circle a^-\-y^ — a = 0, are the only points at which the curve is perpendicular to the axis of x. To find the points at which the curve is parallel to the axis of a;, we must write (ir = 0, this gives (a;' + y' - a) X + 4^ = 0, 374 A MEMOIR UPON CAUSTICS. [145 and thence ic* + y* — a = — 4^ X and ^ + a^ (a? - m) = : this equation will have three real roots if ^<-^=-, and only a single real root if 27 4mv 4m A > -^=- ; for J. = -^ , the equation in question will have a pair of equal roots. It is easy to see that there is always a single real root of the equation which gives rise to a real value of y, Le. to a real point upon the curve; but, when the equation has three real roots, two of the roots may or may not give rise to real points upon the curve. XXXVI. It is now easy to trace the curve. First, when m = 0, or the directrix through the centre of the dirigent circle, the curve is here an oval bent in so as to have double contact with the directrix, and lying on the one or the other side of the directrix according to the sign of A, See fig. a. Fig. a. Fig. h. ..' Fig. c. Fig. d. 145] A MEMOIR UPON CAUSTICS. 375 Next, when the directrix does not pass through the centre of the dirigent circle, it will be convenient to suppose always that m is positive, and to consider A as passing first from to oo and then from to — oo , i. e. to consider first the different inside curves, and then the different outside curves. Suppose a > —^ , the o inside curve is at first an oval, as in fig. 6, where (attending to one side only of the axis) it will be noticed that there are three tangents parallel to the axis, viz. one for the convexity of the oval, and two for the concavity. For A = -^ the two tangents for the concavity come together, and give rise to a stationary tangent (Le. a tangent at an inflection) parallel to the axis, and for ^ > ^ the two tangents for the concavity disappear. The outside curve is an oval (of course on the opposite side of, and) bent in so as to have double contact with the directrix. Next, if a = —^ , the inside curve is at first an oval, as in fig. c, and there are, as before, three tangents parallel to the axis : for A = -^=- , the tangents for the con- cavity of the oval come to coincide with the axis, and are tangents at a cusp, and for A > -g^^ the cusp disappears, and there are not for the concavity of the oval any tangents parallel to the axis. The outside curve is an oval as before, but smaller and more compressed. Next, a < -^ > m*, then the inside curve is at first an oval, as in fig. d, and there are, as before, three tangents parallel to the axis; when A attains a certain 477l' value which is less than y , the curve acquires a double point ; and as A further increases, the curve breaks up into two separate ovals, and there are then only two tangents parallel to the axis, viz. one for the exterior oval and one for the interior oval. As A continues to increase, the interior oval decreases ; and when A attains a certain value which is less than -nfr* ^^^ interior oval reduces itself to a conjugate point, and it afterwards disappears altogether. The outside curve is an oval as before, but smaller and more compressed. Next, if the directrix touch the dirigent circle, i.e. if a = m\ Then the inside curve is at first composed of an exterior oval which touches the dirigent circle, and *)f an interior oval which lies wholly within the dirigent circle. As A increases the interior oval decreases, reduces itself to a conjugate point, and then disappears. The outside curve is an oval which always touches the dirigent circle, at first very small (it may be considered as commencing from a conjugate point corresponding to ^ = 0), but increasing as A increases negatively. 376 A MEMOIR UPON CAUSTICS. [l45 Next, when the directrix does not meet the dirigent circle, i.e. if a < m*. The inside curve consists at first of two ovals, an exterior oval lying without the dirigent circle, and an interior oval lying within the dirigent circle. As A increases the interior oval decreases, reduces itself to a conjugate point and disappeara The outside curve is at first imaginary, but when A attains a sufficiently large negative value, it makes its appearance as a conjugate point, and afterwards becomes an oval which gradually increases. Next, when the dirigent circle reduces itself to a point, i.e. if a = 0. The inside curve makes its appearance as a conjugate point (corresponding to ^ = 0), and as A increases it becomes an oval and continually increases. The outside curve comports itself as in the last preceding case. Finally, when the dirigent circle becomes imaginary, or has for its radius a pure imaginary distance, i.e. if a is negative. The inside curve is at first imaginary, but when A attains a certain value it makes its appearance as a conjugate point, and as A increases becomes an oval and continually increases. The outside curve, as in the preceding two cases, comports itself in a similar manner. The discussion, in the present section, of the different forms of the curve is not a very full one, and a large number of figures would be necessary in order to show completely the transition from one form to another. The forms delineated in the four figures were selected as forms corresponding to imaginary values of the parameters bj means of which the equation of the curve is usually represented, e.g. the equations in Section xxvin. XXXVII. It has been shown that for rays proceeding from a point and refracted at a circle, the secondary caustic is the Cartesian; the caustic itself is therefore the evohite of the Cartesian ; this affords a means of finding the tangential equation of the caustic. In fact, the equation of the Cartesian is {x'-i-y'- ay + 16^ (a? - m) = ; and if we take for the equation of the normal Zf+F7; + ^ = 0, (where f, t) are current coordinates), then : a? (ar* -f y^ - a) H- 4.4 : 4ily, 145] A MEMOIR UPON CAUSTICS. 377 equations which give Z*7» (a;» + y» - a) = ^AZ'XY^ whence eliminating, we have {Z» + X (mZ' - AX*)Y + 7» (mZ^ - AXy - Z»P (oZ + 4ilZ) = 0, where if, as before, c denotes the radius of the refracting circle, a the distance of the radiant point from the centre, and fi the index of refraction, we have A = d'a The above equation is the condition in order that the line Xx + Fy H- Z = may be a normal to the secondary caustic (a;" + y' — a)'+16^ (a? — m) = 0, or it is the tangential equation of the caustic, which is therefore a curve of the class 6 only. The equation may be written in the more convenient form xxxvin. To compare the last result with that previously obtained for the caustic by reflexion, I write ft = — 1, and putting also c = l and Z = a (for the equation of the reflected ray was assumed to be Xx + Fy + a = 0), we have a = a« + 2, A^\a, m = 2^(l + 2a«), a,nd the equation becomes, after a slight reduction, * 4a< + 4o'Z (2o» + 1 - Z») + (Z» + F») (2o» + 1 - X")" - 4ia?Y* (a« + 2 + IX) = 0, ^which may be writteu (2o' + Z (2o« + 1 - Z»))« + F» ( - 40' + 1 - Sa'X - 2 (2a'' + 1) X' + X*) = ; this divides out by the fector (X + iy, and the equation then becomes, (Z» - Z - 2a')> + F» ((Z - ly - 4a') = 0, ■w-hich agrees with the result before obtained. 378 A MEMOIR UPON CAUSTICS. [145 XXXIX. Again, to compare the general equation with that previously obtained for parallel rays refracted at a circle, we must write /a = t> c=1, a = x, Z=^k (for the equation of the refracted ray was taken to be Xx -f Yy + ^ = 0) ; we have then a^l-vk^-Vk'a^ A^^k'a^ m = ^ (l +(1 + i»)a*) , and, after the substitution, a = x . The equation becomes in the first instance Jfc« + 2*»Z |~ (!+(!+ k") a«) t« - i**az4 + {X' + F«) — (l + (1 + i») a«) k' - J**(f.Y«J' and then putting a = x , or, what is the same thing, attending only to the tenuN which involve a', and throwing out the constant factor k^, we obtain (X^ + F«)(.Y« - 1 - k'Y - 4^'«F* = 0, or x^(X^-i-i(f^y+Y^{X-}-i+k)(X-i''k)(x + i-k){X--i-k)-~ 0, which agrees with the former result. XL. It was remarked that the ordinary construction for the secondary caustic* could not be applied to the case of parallel rays (the entire curve would in fact pass off to an infinite distance), and that the simplest course was to measure the distance GQ from a line through the centre of the refracting circle perpendicular to tin direction of the rays. To find the ecjuation of the resulting curve, take the centre of the circle as the origin and the direction of the incident rays for the axis of jc\ let the radius of the circle be taken equal to unity, and let fi denote, as before, the index of refraction. Then if a, /3 arc the coordinates of the point of incidence of a ray, we have a'^ + y^=l. and considering a, )8 as variable parameters connected bv this equation, the retjuired curve is the envelope of the circle, Write now a = cos^, y8 = sin^, then multiplying the equation by —2, and writing 1 + cos 20 instead of 2 cos^ 9, the equation becomes 1 + cos 20 - 2fi^ (x' -b if - 2x cos - 2i/ sin 5 + 1) = 0, which is of the form ^cos2^ + 5sin2(9 + C'cose + i)sin^ + A' = 0, 145] A MEMOIB UPON CAUSTICS. 379 and the values of the coefficients are 4 = 1, B = 0, c = v^. D = Vy. E = -2fi*(ai' + y*) - 2yit»+ 1. Substituting these values in the equation 12 (il» + £») - 3 (C» + i)*) + 4£>1' - {27il (C* - D") + BiBCD - (72 (4« + £«) + 9 (C* + i)»)) JE^ + 8£>}' = 0, the equation of the envelope is found to be 16 {(1 - /*» + /*«) - 0*' + ,*«) {a? + r/») + At* (a^ + y')*;' ( 4 - 6/*' - 6/** + 4|t« - (6/** + 3/** + Qfif) (a?+y>)- 27/*« (a? - yO -(6/*« + 6At')(«' + 2/')' ^ > S-=o, which is readily seen to be only of the 8tb order. But to simplify the result, write iirst (a? + y*-l) + l, and 2*»-l -(«' + y*-l) in the place of a? + y* and a?-f respec- tively, the equation becomes 4{(1 -/*')»- ft* (1 -/*») (a;* + y» - 1) + /** (a;* + y* - 1)"}* ' 2(1 -/*•)» - 3/*" (1 - A*')* («^ + y* - 1) - 27/**a? — -< \ -3M*(i-/*»)(a!?+y-i)» + 2/*« (a^ + y» - 1)» >• =0. / Write for a moment 1— /a' = 5, ft*(.'c* + y'— l) = p, the equation becomes or developing, + 54 (2g» - 39«p - Sgp* + 2/o») ;A*a;« - 72 Var* = 0, and reducing and dividing out by 27, this gives 9«P»(P-5)» + 2(p + })(2p-})(p-23);aV-27;aV = 0, 48—2 380 A MEMOIR UPON CAUSTICS. [145 whence replacing q, p by their values, the required equation is + 2 (/x« (a;* 4- ya) - 2;a« + 1) (2/^2 (a;» + y*) - /*« - 1) (/*« (a;« + y«) - 2 + ;a«) a;^ which is the equation of an orthogonal trajectory of the refracted rays. In the case of reflexion, ;a = — I, and the equation becomes 4(a;« + y*-l)»-27a^=0. Comparing this with the equation of the caustic, it is easy to see, Theorem. In the case of parallel rays and a reflecting circle, there is a secondaT"^ caustic which is a curve similar to and double the magnitude of the caustic, t\ position of the two curves differing by a right angle. XLI. The entire system of the orthogonal trajectories of the refracted rays might like manner be determined by finding the envelope of the circle (where, as befo: a, )8 are variable parameters connected by the equation a"-f )9* = 1), {The result, as far as I have worked it out, is as follows, viz. — (3 - 12 [m^ + 'Zmfi^x + ;a* (a:* + y')] + [1 - 2;a« + 2m* - 2/x« {a^ + y»)]*)» - ([1 - 2/A« + 2m» - 2/x» (a^ + f)] [9 + 18ah« + 36m/A«a: + 18;a* {a^ -h f)] - 54 [m» + imfi^x -^ti^ia^- /)] - [1 - 2;a« + 2iii» - 2;a« {a^ + y»)]»)* = 0, which, it is easy to see, is an equation of the order 8 only. Added Sept. 12. — A. C.j 146. A MEMOIK ON CUKVES OF THE THIRD ORDER. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 415 — 446. Received October 30, — Read December 11, 1856.] A CURVE of the third order, or cubic curve, is the locus represented by an equation such as U={*^x, y, ^)» = 0; and it appears by my "Third Memoir on Quantics," [144], that it is proper to consider, in connexion with the curve of the third order 17=0, and its Hessian J? 17=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of J7, and therefore, when the variables x, y, z o{ U are considered as point coordinates, the variables f, 17, (f of PJ7 and QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. [The curve PU=0 is now usually called the Cayleyan.] A geometrical definition of the Pippian was readily found; the curve is in fact Steiner's curve i2o mentioned in the memoir "AUgemeine Eigenschafben der algebraischen Curven," Crelle, t. XLVii. [1854] pp. 1 — 6, in the particular case of a basis-curve of the third order; and I also found that the Pippian might be considered as occurring implicitly in my "M^moire sur les courbes du troisifeme ordre," Liouville, t. IX. [1844] pp. 285 — 293 [26] and "Nouvelles remarques sur les courbes du troisi^me ordre," Liouville, t. X. [1845] pp. 102 — 109 [27]. As regards the Quippian, I have not succeeded in obtaining a satisfactory geometrical definition ; but the search aft^r it led to a variety of theorems, relating chiefly to the first-mentioned curve, and the results of the investi- gation are contained in the present memoir. Some of these results are due to Mr Salmon, with whom I was in correspondence on the subject. The character of the results makes it difficult to develope them in a systematic order; but the results are given in such connexion one with another as I have been able to present them 382 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 in. Considering the object of the memoir to be the establishment of a distinct geometrical theory of the Pippian, the leading results will be found summed up in the nine different definitions or modes of generation of the Pippian, given in the con- cluding number. In the course of the memoir I give some further developments relating to the theory in the memoirs in Liouville above referred to, showing its relation to the Pippian, and the analogy with theorems of Hesse in relation to the Hessian. Article No. 1. — Definitions, <tc. 1. It may be convenient to premise as follows: — Considering, in connexion with a curve of the third order or cubic, a point, we have : (a) The first or conic polar of the point. (b) The second or line polar of the point. The meaning of these terms is well known, and they require no explanation. Next, considering, in connexion with the cubic, a line — (c) The first or conic polars of each point of the line meet in four points, which are the four poles of the line. {d) The second or line polars of each point of the line envelope a conic, which is the lineO'polar envelope of the line. And reciprocally considering, in connexion with a curve of the third class, a line^ we have: {e) The first or conic pole of th^ line. (/) The second or point-pole of the line. And considering, in connexion with the curve of the third class, a point — (g) The first or conic poles of each line through the point touch four linefi, which are the four polars of the point. (h) The second or point poles of each line through the point generate a conic which is the point-pole lociis of the point. But I shall not have occasion in the present memoir to speak of these reciprocal figures, except indeed the first or conic pole of the line. The term conjugate poles of a cubic is used to denote two points, such that the first or conic polar of either of them, with respect to the cubic, is a pair of lines passing through the other of them. Reciprocally, the term conjugate polars of a curve of the third class denotes two lines, such that the first or conic piole of either of them, with respect to the curve of the third class, is a pair of points lying in the other of them. 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 383 The expression, a ayzygetic cubic, used in reference to two cubics, denotes a curve of the third order passing through the points of intersection of the two cubics; but in the present memoir the expression is in general used in reference to a single cubic, to denote a curve of the third order passing through the points of intersection of the cubic and its Hessian. As regards curves of the third class, I use in the memoir the full expression, a curve of the third class syzygetically connected with two given curves of the third class. It is a well-known theorem, that if at the points of intersection of a given line with a given cubic tangents are drnwn to the cubic, these tangents again meet the cubic in three points which lie in a line; such line is in the present memoir termed the satellite line of the given line, and the point of intersection of the two lines is termed the satellite point of the given line; the given line in reference to its satellite line or point is termed the primary line. In particular, if the primary line be a tangent of the cubic, the satellite line coincides with the primary line, and the satellite point is the point of simple inter- section of the primary line and the cubic. Article No. 2. — Oroup of Theorems relating to the Conjugal Poles of a Vubic, 2. The theorems which I have first to mention relate to or originate out of the theory of the conjugate poles of a cubic, and may be conveniently connected together and explained by means of the accompanying figure. The point ^ is a point of the Hessian ; this being so, its first or c(mic polar, with respect to the cubic, will be a pair of lines passing through a point F of the Hessian ; and not only so, but the first or conic polar of the point F, with respect to the cubic will be a pair of lines passing through E. The pair of lines through 384 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 F are represented in the figure by FBA, FDC, and the pair of lines through E are represented by EC A, EDC, and the lines of the one pair meet the lines of the other pair in the points A, B, C^ D. The point 0, which is the intersection of the lines AD, BC, is a point of the Hessian, and joining EOy FO, these lines are tangents to the Hessian at the points E, F, that is, the points E, F are corresponding points of the Hessian, in the sense that the tangents to the Hessian at these points meet in a point of the Hessian. The two points E, F are, according to a preceding definition, conjugate poles of the cubic. The line EF meets the Hessian in a third point 0, and the points G, are conjugate poles of the cubic. The first or conic polar of 0, with respect to the cubic, is the pair of lines AOD, BOC meeting in 0. The first or conic polar of 0, with respect to the cubic, is the pair of lines QEF and Gfeff! meeting in C The four poles of the line EO, with respect to the cubic, are the points of intersection of the first or conic polars of the two points E and 0, that is, the four poles in question are the points jP, F, e, e'. Similarly, the four poles of the line FO, with respect to the cubic, are the points E, E, /, f. The line EF, that is, any line joining two conjugate poles of the cubic, is a tangent to the Pippian, and the point of contact V is the harmonic with respect to the points E, F (which are points on the Hessian) of G, the third point of intersection with the Hessian. Conversely, any tangent of the Pippian meets the Hessian in three points, two of which are conjugate poles of the cubic, and the point of contact is the harmonic, with respect to these two points, of the third point of intersection with the Hessian. The line GO in the figure is of course also a tangent of the Pippian, and more- over the lines FBA, FDC (that is, the pair of lines which are the first or conic polar of E) and the lines EGA, EDB (that is, the pair of lines which are the first or conic polar of F) are also tangents to the Pippian. The point E represents any point of the Hessian, and the three tangents through E to the Pippian are the line EFG and the lines EGA, EDB; the line EFG is the line joining E with the conjugate pole F, and the lines EGA, EDB are the first or conic polar of this conjugate pole F with respect to the cubic. The figure shows that the line EG (the tangent to the Hessian at the point E) and the before-mentioned three lines (the tangents through E to the Pippian), are harmonically related, viz. the line EG the tangent of the Hessian, and the line EF one of the tangents to the Pippian, are harmonics with respect to the other two tangents to the Pippian. It is obvious that the tangents to the Pippian through the point F are in like manner the line GFE, and the pair of lines FBA, FBG, and that these lines are harmonically related to FO the tangent at F of the Hessian. And similarly, the tangents to the Pippian through the point are the line GO and the lines AOD, BOO, and the tangents to the Pippian through the point G are the line GO and the lines GFE and Gfefe', Thus all the lines of the figure are tangents to the Pippian except the lines EG, FO, which are tangents to the Hessian. It may be added, that the lineo-polar envelope of the line EF with respect to the cubic is the pair of lines OE, OF, 146 J A MEMOIR ON CURVES OF THE THIRD ORDER. 385 It will be presently seen that the analytical theory leads to the consideration of a line IJ (not represented in the figure): the line in question is the polar of E (or F) with respect to the conic which is the first or conic polar of F (or E) with respect to any syzygetic cubic. The line /J" is a tangent of the Pippian, and more- over the lines EF and // are conjugate polars of a curve of the , third class syzygetically connected with the Pippian and Quippian, and which is moreover such that its Hessian is the Pippian. Article Nos. 3 to 19. — Analytical investigations, comprising the proof of the theoremSy Article No. 2. 3. The analytical theory possesses considerable interest. Take as the equation of the cubic, Z7 = ir' + y* + '^*+ ^locyz = ; then the equation of the Hessian is HU= f(af + f + z')-(\ + W) xyz^O; and the equation of the Pippian in line coordinates (that is, the equation which expresses that ^x+r)y-^ ^z = is a tangent of the curve) is Pir= - Z(|» -h 7;»+ ?») + (- 1 + 4P) ?i7?= 0. The equation of the Quippian in line coordinates is Qir= (1 - lOP) (f -h 7;» + ?») - 61^ (5 + 4P) fi;C= ; and the values of the two invariants of the cubic form are T=1-20P-8Z«, values which give identically, 2^-64fll* = (l+8Z'')»; the last-mentioned function being in fact the discriminant. 4. Suppose now that (X, Y, Z) are the coordinates of the point E, and (X\ Y, Z') the coordinates of the point F; then the equations which express that these points are conjugate poles of the cubic, are XT-{-l(YZ'^-rZ) =0, 7F-hi(ZX'+^Z) = 0, ZZ' +1{XY + X'Y)=0; and by eliminating from these equations, first (X\ Y, Z\ and then (X, F, Z), we find ZH^* + F» + Z»)-(l+2P)XFir =0, p (X» + F» + Z'O - (1 + 2P) X'YZ' = 0, which shows that the points Ey F are each of them points of the Hessian. C. II. 49 386 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 5. I may notice, in psussing, that the preceding equations give rise to a somewhat singular imaymmetrical quadratic transformation of a cubic form. In hct, the second and third equations give X' : T : Z'^YZ-PX^ : PX7-l^ : PZX-lYK And sub- stituting these values for X\ Y, Z' in the form Z«(Z'»+ F» + Z'»)-(l + 2l')X'rZ', the result must contain as a factor f(Z»+ F» + Z») - (1 + 2P) XTZ\ the other &ctor is easily found to be -P(P(Z«+F» + 2*) + 3ZZFZ). Several of the formulsB given in the sequel conduct in like manner to unsymmetrical transformations of a cubic form. 6. I remark also, that the last-mentioned system of equations gives, symmetricaUy, X'* : y : Z" : FZ' : Z'X' : X'T = YZ-l*X* : ZX-l'Y^ : XT-PZ* : 1*YZ-IX* : 1*ZX-IY* : PXY-W; and it is, I think, worth showing how, by means of these relations, we pass from the equation between X', Y', Z! to that between X, Y, Z. In fact, representing, for shortness, the foregoing relations by X'* : T* : Z* : TZ' : Z'X' : X'T ^ A . B : G : F : Q : H, we may write X' = AF=GH, T = BQ = HF, Z' = CH = FQ, ABC = FGH; and thence X'*='AF.G'H\ T* = BO.H*F^, Z'* = CH.F>G', X'TZ^F'O'H*; hence l^iX'* + T* + Z*)-(l+2l*)X'TZ' = FGH{l^iAQH+BHF+CFQ)-(l + 2i')FOH}. But we have l^(AOH + BHF+ CFG) = - (21' + 1») (X* + F* + Z») XYZ+{1* + 2^)(Y*Z* + Z'X* + X*Y*), -(l + 2lf)FGH = (^ + 2l!>)(X*+Y* + Z*)XYZ+(l*+2P)(Y*Z* + Z*X* + X*Y') + l'(l-l^)(l + 2l*)X'Y*Z'; and thence P (AGH+BHF+ CFG) - (1 + 21') FGH = _ p(l _ P) {P(Z«+ Y*+Z*)XYZ- (1 + 2l^)X'Y*Z'} ; and finally, l'(X"+ Y'* + Z'')-(l + 2l')X'Y'Z' = l'{-l + l*)(lYZ-X'){lZX- Y*)(IXY-Z')XYZ >i{l*(X'+Y*-\-Z*)-{l+2fi)XYZ\. 146] A MEMOnt ON CURVES OF THE THIRD ORDER. 387 We have also, identically, ABG-FQH = j(-l+l*)XYZ{l*(X*-\-Y* + Z>)-(l + 2l*)XYZ}, which agrees with the relation ABG — FGH=0. 7. Before going further, it will be convenient to investigate certain relations which exist between the quantities (X, Y, Z), (X', Y, Z), connected as before by the equations XX' + liYZ' +TZ) =0, YT + 1 {ZX' + Z'X) = 0, ZZ' +i(XF' + Z'F) = 0, and the quantities ^=YZ'-TZ, a=XX' = -j(YZ + TZ), v=zx'-z'x, fi=Yr=-jizx'+z'X), r = Zr - Z'F, y=ZZ' =-- (XT + X'Y). We have identically, 2XX'(YZ'-TZ) + (XT + X'Y)(ZX'-ZX) + {ZX' + Z'X)(XT-X'Y) = 0; or expressing in terms of f, 17, ^, a, /9, 7 the quantities which enter into this equation, and forming the analogous equations, we have 2K- fV- /9r=0, (A) -y^+2lfir,- ar=0, -/S|- o,, + 2tyi:=0. We have also X*rZ-X''YZ = i{-(XT + X'Y)(ZX'-Z'X) + {ZX' + Z'X)(XY'-X'Y)}, and thence in like manner, X»rZ' - X"YZ = ^ (7^ - /30, (B) Z*X'r - X'*YZ = i (j8f - ar,l Again, we have ( YZ' - rzy ={YZ'+ YZf - 4 ytzz', (ZX' -Z'X)(Xr - X'Y)^-(ZX' + ZX)(Xr + X'Y) + 2XX' (YZ' + TZ); 49—2 388 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 and thence p= i«'-4/87, (C) 1;*= |-,/3»-4ya, and conversely J(l + 8P)«'= ?-4?.;r. (D) -l(l + 8P)7a=2V+ ?f. -J,(l + 8P)a/9 = 2ZC»+ f,. 8. It is obvious that ix + fiy + J^ = is the equation of the line EF joining the two conjugate poles, and it may shown that aa? + /3y + 7-? = is the equation of the line /J", which is the polar of E with respect to a con ^ which is the first or conic polar of F with respect to any syzygetic cubic. In fitc^ the equation of a syzygetic cubic will be aj' + y*+^H-6Xa;y-? = 0, where X is arbitrar]!^ and the equation of the line in question is or developing, XTx+YYy + ZZ'z •\-\{YZ''^YZ)x-\-{ZT^Z'X)y + {XT + TY)z]^0\ 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 389 and the function on the left-hand side is which proves the theorem. 9. The equations (A) by the elimination of (f, % f), give -Z(a» + /S»+7») + (-H-4Z») 0^7=0, which shows that the line IJ is a tangent of the Kppian : the proof of the theorem is given in this place because the relation just obtained between a, /9, 7 is required for the proof of some of the other theorems. 10. To find the coordinates of the point in which the line EF joining two conjugate poles again meets the Hessian. We may take for the coordinates of G, and, substituting in the equation of the Hessian, the terms containing u^ 1^ disappear, and the ratio u \ v \b determined by a simple equation. It thus appears that we may write V = 3P(Z»Z' -f Y^T + Z^Z')-{\ -f 2P) {YZT + ZXY -h XYZ') ; hence introducing, as before, the quantities f, 17, f, a, /9, 7, we find uZ + vX = 3P (717 - )80 + (1 + 2P) (-ST'FZ' - Z'«F^ ; but from the first of the equations (B), and therefore the preceding value of uX-^-vX' becomes which is equal to —21— (y't - ^^^- Hence throwing out the constant factor, we find, for the coordinates of the point (?, the values 71; -^f, af-7f, ^f-ai7. 11. To find the coordinates of the point 0. Consider as the point of intersection of the tangents to the Hessian at the points E, Fy then the coordinates of are proportional to the terms of 3PZ» - l-f- 2PFZ , ZM^ - l-f- n^ZX , 3?2? - 1 + 2PZF 3PZ'«-H-2Z»F^', 3PF«-1-h2Z»Z'Z', 3PZ'«- 1 -f 2PZ'F' 390 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 Hence the a;-coordinate is proportional to which is equal to 9i*(F»Z'«-F«2?)-l-3P(l + 2P)FF(XF-Z'r) + 3P(l + 2P)ZZ'(^Z'-Z'Z) -(1 + 2P)» ZZ'(FZ'-rZ); or introducing, as before, the quantities f, 17, f, a, /3, 7, to - 9Paf + 3P (1 + 2P) (/9?+ w) - (1 + Wy of, = (- 1 - 13/» - «•) af + 2Z« (1 + 2Z») (/9? + 71;). But by the first of the equations (A) /3f + 71; = 2Zaf , and the preceding value thus becomes ( — 1 — 7P + 8P) af. Hence throwing out the constant factor the coordinates of the point are found to be af fiVy 7?- 12. The points (?, are conjugate poles of the cubic. Take a, 6, c for the coordinates of G, and a', h\ c* for the coordinates of 0, we have a, 6, c =r7-^?, a?-7f /3f-a^, a', 6', C = af , /9i7 , 7?. These values give aa' + i (6c' + 6'c) = «f(r7-/8?) + M/8'7(/8f-«'7) + 7?K-7f)} = ^7(«7 + W + '7"(~^«^) + r*(M + f?(-«/8-V); or substituting for f?/, 17', f ^ f f their values in terms of a, /9, 7, this is + ( f -47a)( -/a/8) + ( ^ -4ay9)( /a7) + (-f/9«-ia7)(~«/8-V), which is identically equal to zero. Hence, completing the sjrstem, we find aa' + i (6c' + 6'c) = 0, W -h I {ca' + c'a) = 0, co' + i (a6' + a'6) = 0, equations which show that (as well as G) is a point of the Hessian, and that the points Qy are corresponding poles of the cubic. 146] A MEMOm ON CUBVES OF THE THIRD OBDEB. 391 13. The line EF joining a pair of conjugate poles of the cubic is a tangent of the Pippian^ In fact, the equations (A), by the elimination of a, /8, 7, give -f(P + ^*+?»)-l-(-l+4P)^7?=0, which proves the theorem. 14. To find the equation of the pair of lines through F, and to show that these lines are tangents of the Pippian. The equation of the pair of lines considered as the first or conic polar of the conjugate pole E, is Z (a;» + nyz) + Y{y^ + 2lzx) + Z(z* + 2lxy) = 0. Let one of the lines be Xa? -f A^y + v^ = 0, then the other is X Y Z ^ and we find 2lXfiv - Fi/» - Z/i« = 0, - Xv" + 2lYv\ - ^« = 0, - Z/A« - yx« + 2iZfiv = 0, any two of which determine the ratios X, /a, 1/. The elimination of Z, F, Z gives 2Z/A1;, -i/», - /A« =0, - v" , 2,lv\ - X« - /A« , - XS 2iX/i which is equivalent to X/ii; { -Z(X» + /i« + !;») + (- 1 + 4P) X/ii/} = 0; or, omitting a factor, to - /(X» + /i« + !/») + (- 1 +4i»)X/ii; = 0, which shows that the line in question is a tangent of the Pippian. 15. To find the equation of the pair of lines through 0, The equation of the pair of lines through ^ is in like manner Z'(a;* + 2iy2r)+ F(y« + 2i«a?) + Z'(^« + 2Ziry) = 0; ^ Steiner's onrre Rf^ in the partioalar oase of a onbio baais-onrve, is aooording to definition tiie envelope of the line EF^ that is, the onrve R^ in the partionlar case in question is the Pippian. 392 A MEMOIIt ON CURVES OF THE THIRD ORDER. [l46 and combining this with the foregoing equation, X{a? + 2lyz) -f F(y' -f 2lzx) + Z(z^ + 2lxy) = of the pair of lines through F, viz. multiplying the two equations by X^X' -h PF 4- Z^Z\ - {XX'^ + YT^ + ZZ'% and adding, then if as before we find as the equation of a conic passing through the points A, B, C, D, the equation a (ic» + 2lyz) + h{y^ + 2lzx) + c{z'^ + 2lxy) = 0. But putting, as before, then a', 6', c' are the coordinates of the point 0, and the equations aa' + i(6c' + 6'c) = 0, 66' + /(ca'+c'a) = 0, show that the conic in question is in fact the pair of lines through the point 0. 16. To find the coordinates of the point F, which is the harmonic of with respect to the points E, F. The coordinates of the point in question are uX-vT, uY-vT, uZ-vZ', where u, v have the values given in No. 10, viz. u = -3i»(ZZ'«+ 77'« + ZZ'0 + (l + 2P)(FZ'X + ^Z'F + X'rZ), V = 3Z« (Z»X' + PF' + Z^Z') - (1 + 2P) (YZX' + ZXY' + XYZ") ; these values give uX - vZ' = - 81' {2Z»Z'« + (Zr + Z' F) YY' + (ZZ' + Z'Z) ZZ] + (l + 2l^){(XY' + X'Y)(XZ'-\-X'Z) + Xr(YZ'+Y'Z]] and therefore iiZ-t;Z' = -3p|2a«-?/37 ■f(l + 2i«)|iy97-Ja«. = J,(l+8?)(-Za> + /87); and consequently, omitting the constant factor, the coordinates of F may be taken to be -fa« + /97, -//3» + 7a, -V + o^- 146] A MEMOIR ON CURVES OP THE THIRD ORDER. 393 17. The line through two consecutive positions of the point F is the line EF. The coordinates of the point T are -•la^-^l3y, ^l^ + ya, ^Irf + a/S; and it has been shown that the quantities a, /9, 7 satisfy the equation - Z (a» -f i8» 4- 7») + (- 1 + 4P) a/97 = 0. Hence, considering a, /9, 7 as variable parameters connected by this equation, the equation of the line through two consecutive positions of the point F is I ^si(^ + {-l + 4l^)fiy, -3//S«-l-(-l+4P)7a, - SZ^-h (- 1 -h 4P) a/3 > = ; ^, -2la , 7 , /3 y, 7 , -2Z/9 . a z, /3 , OL , -2/7 and representing this equation by Lx + My -{• Nz = 0, we find Z= (4P/37 - a») (- 3/a» + (- l-f- 4/0/87) + (a/9+ 2/7') (- n^ + (- 1 + 4P) 7a) + (07 + 2/^) (- 3^y^ +(-l-h4P)a/9); or, multiplying out and collecting, L = 3Za* -h (- 1 - 8/») a?Py -h (- 5/ + 8P) (a/8» -h a7») -f- (- 16/» + 16/») /3V ; but the equation - Z (a» -f- y3» -f- y) + ( - 1 + 4/^ 0/87 = gives 3^0* = - 3i (ay8« + OT*) + (- 3 + 12i») tf»/97, and we have Z = (- 4 + 4P) OL^Py + (- 8i + SV) (o/S^-h ay) + (- 16P + 16Z0/3V = (- 4 + 4Z») (a»/37 + 2Z (a5» -h a^) + 4Z»/8V) = (_ 4 + 4i.) (a^ + 2/^) (a/9 -h 2ty») ; or, in virtue of the equations (D), Z = (-4 + 4P)Z«5f .Pfi7 = (-4 + 4P)i*fi7?=(-4-h4/»)/*fi;?.f. Hence, omitting the common factor, we find L : M : N=^ : r) : ^, and the equation Lx + My + Nz = becomes f a? + w + C^ = 0, C. II. 50 394 A MEMOIB ON CUBYES OF THE THIRD ORDER. [l46 which is the equation of the line EF, that is, the line through two consecutive positioDs of r is the line EF\ or what is the same thing, the line EF touches the Pippian in the point F which is the harmonic of Q with respect to the points E, F. 18. The lineo-polar envelope of the line EF, with respect to the cubic, is the pair of lines OE, OF. The equation of the pair of lines OE, OF, considered as the tangents to the Hessian at the points E, F, is r=0. X {(3PZ'« - 1 + 2PFZ0 X -f- {ZP r* - 1 + WZ'X') y + (3P^'* - 1 + 2PX'Y') z] ) Here on the left-hand side the coefficient of a;* is 9/*Z«Z'« - 3? (l-f- 2V) (X'Y'Z' + X'^YZ) + (1 + 2P)« YTZ^, which is equal to that is 91^ - 3P (1 + 2Z») (Z«/97 + ^ a«) + (1 -h 2Z»)» /Sy, j(-/ + ?){3ia»+2(l-h2P)/37}; and the coefficient of yz is 9^(F«Z'»+ F«2?) -3Z«(1 + 2V) {YY' {XY' + X'Y)^ ZZ'iXZ' -^^X'Z)) -h(l+2P)«ZZ'(F^+F'Z), which is equal to that is J(-/-l-/*){(l-4/»)a«-6P/97) Hence completing the system and throwing out the constant factor, the equation of the pair of lines is (3k«-|-2(H-2Z»)/87, 3/i8« + 2 (1 + 2P) 7a, 3^ + 2 (1 -h 2Z') a/9. (1 » 4P) a« - 6/^/87, (1 - 4P) y3« - QPyOL, (1 - 4P) 7* - &Pol^\x, y, ^)« = 0. But the equation of the line EF is |a? -h ^y + f^ = 0, and the equation of its liueo-polar envelope is f, X, Iz. ly f), Iz, y, Ix ?, ly, X , z = 0; 146] A MEMOIR ON CUBVES OF THE THIBD OBDEB. 395 or expanding, or arranging in powers of x, y, z, (-/«p-2/i7r, -tv-m. -^"^-2i?i7, ip+^;?, hv'+m> iP+mii^,y>^y^o: and if in this equation we replace f , &c. by their values in terms of a, fi, y, aa given by the equations (D), we obtain the equation given as that of the pair of lines OE, OF. 19. It remains to prove the theorem with respect to the connexion of the lines EF, TJ. The equations (A) show that the two lines ^x+Tfy +5e^ = 0, ax-\-fiy+^=0, (where f, i;, f and a, /3, y have the values before attributed to them) are conjugate polars with respect to the curve of the third class, in which equation |; 17, ^ denote current line coordinates. The curve in question is of the form APV +BQV = 0. We have, in fact, identically, It is clear that the curve in question must have the curve PCr = for its Hessian; and in fact, in the formula of my Third Memoir, [144] H{6aPU + fiQU)=^(-2T, 48S*, ISTS , T*+16S»$a, fiyPU + ( 8>8f, r,-8S*, ^T8 Ha, fiYQU, the coefficient of QCT is and therefore, putting a = JT, ^ = — 4/S, we find JT(3r.Pcr-4S.<2J7) = -H3^-645*)»Pcr. Article No. 20. — Theorem relating to the curve of the third doss, mentioned in the preceding Article, 20. The consideration of the curve ST.PU— 4aS.QU=0, gives rise to another geometrical theorem. Suppose that the line (^, 17, (f), that is, the line whose equation is ^ + rfy + ^z = 0, is with respect to this curve of the third class one of the four polars of a point {X, T, Z) of the Hessian, and that it is required to find the envelope of the line ^x + r)y+^z = 0, 50—2 396 A UEHOIB ON CURVES OP THE THIRD ORDER. [U6 We have X: y,^-ip-,j;: V-ff^if-ft, and X, y, Z are to be eliminated from these equations, and the equation PiX*+V' + Z*)-{l+2l')XrZ=0 of the Hessian. We have x'+Y'+z'= Hp+v'+^y + 91 fVf -(l+2P)(i,'f + rP + ?-;*). xYz= i(p+y+c.)f,{: 4 (-1+1')^^ -^(VS' + rP + P'j'). and thence nu= P (^+^> + ^y -(i + 5l^)(P + v' + n^? + (I + 101' -21*)^^; and equating the right-hand side to zero, we have the equation in line coordinates of the curve in question, which is therefore a curve of the sixth class in quadratic syzygy with the Pippian and Quippian. Articl« No. 21. — Geometrical definition of the Quippian, 21. I have not succeeded in obtaining any good geometrical definition of the Quippian, and the following is only given for want of something better. T. PU \PeH(aU + 6^HU)\ - P{6HU) [T{aU+&^HU) .P{aU+ Q^HU)] =0, which is derived in what may be taken to be a known manner from the cubic, is ii general a curve of the sixth class. But if the syzygetic cubic aU + 60HV = be properly selected, viz. if this curve be such that its Hessian breaks up into three lines, then both the Pippian of the cubic aU+6^HU = 0, and the Pippian of iU Hessian will break up into the same three points, which will be a portion of the curve of the sixth class, and discarding these three points the curve will sink down to one of the third class, and will in fact be the Quippian of the cubic. To show this we may take an + 60HU = iX!'+y' + A=O 146] A MEMOIIt ON CURVES OF THE THIRD ORDER. 397 as the equation of the syzygetic cubic satisfying the prescribed condition, for this value in fact gives H (aU-^efiHU) = - xyz, = 0, a system of three lines. We find, moreover, P(aU + 6fiHU) = P(x' + f+z'l:=''^^ and P {6H{aU-h 6l3HU)]^P{''6ayz), = -4,^^, the latter equation being obtained by first neglecting all but the highest power of I in the expression of PU, and then writing l = —l: we have also T(aU-\-6l3HU)=l. Substituting the above values, the curve of the sixth class is ^^^^T.PU+P(6HU)]=0', or throwing out the factor fi/Jf, we have the curve of the third class, -^T.PU+P(6HU) = 0. Now the general expression in my Third Memoir, viz. P {aU+ 60HU) = (a» + USafi' + *T^) PU -¥ (c^fi - 48/3*) QU, putting a = 0, /8 = 1, gives P (6HU) = iT.PU-i8.QU, or what is the same thing, -4T.PU + P{6HU) = -4>8,QU; and the curve of the third class is therefore the Quippian QU = 0, It may be remarked, that for a cubic Cr = the Hessian of which breaks up into three lines, the above investigation shows that we have Pf/^= — fi?^, P(6-ffJ7) = — 4fi7f. and T=l, and conse- quently that --^T.PU -{- P{6HU) ought to vanish identically; this in fact happens in virtue of the factor 8 on the right-hand side, the invariant iS of a cubic of the form in question being equal to zero ; the appearance of the factor 8 on the right-hand side is thus accounted for d priori. Article No. 22. — Theorem relating to a line which meets three given conies in six points in involution. 22. The envelope of a line which meets three given conies, the first or conic polars of any three points with respect to the cubic, in six points in involution, is the Pippian. It is readily seen that if the theorem is true with respect to the three conies, dx ' dy * dz * 398 A M£MOIB ON CURVES OF THE THIBD ORDER. [l46 it is true with respect to any three conies whatever of the form ^ dU , dU dU ^ that is, with respect to any three conies, each of them the first or conic polar of some point (X, /a, v) with respect to the cubic. Considering then these three conies, take ^x+riy-\' ^z = as the equation of the line, and let (Z, F, ^ be the coordinates of a point of intersection with the first conic, we have fZ-fi7F+?Z=0, X^+21YZ =0; and combining with these a linear equation aX-{-l37+yZ=0, in which (a, /3, 7) are arbitrary quantities, we have and hence an equation in (a, /3, 7) which is in fact the equation in line coordinates of the two points of intersection with the first conic. Developing and forming the analogous equations, we find ( V' > r , -2Za ^rf , Iv^ , -fiy-if'ia, A 7)" = 0, which are respectively the equations in line coordinates of the three pairs of intersectionB. Now combining these equations with the equation 7 = 0, we have the equations of the pairs of lines joining the points of intersection with the point (« = 0, y = OX and if the six points are in involution, the six lines must also be in involution, or the condition for the involution of the six points is -2^r, ^ , m =0. that is, 4P|:«^; (- f7 - ^D + Vr + ^f?* + 2i»|Vr + Sf'lVC + C* (- f^ - ^D = ; or, reducing and throwing out the fsictor ^, we find -K?+v + r)+(-n-*i')fv?=o, which shows that the line in question is a tangent of the Pippian. 146] A M£MOIB ON CURVES OF THE THIRD ORDER. 399 It is to be remarked that any three conies whatever may be considered as the first or conic polars of three properly selected points with respect to a properly selected cubic curve. The theorem applies therefore to any three conies whatever, but in this case the cubic curve is not given, and the Kppian therefore stands merely for a curve of the third class, and the theorem is as follows, viz. the envelope of a line which meets any three conies in six points in involution, is a curve of the third class. Article No. 23. — Completion of the theory in Liouville, and comparison ivith analogous theorems of Hesse. In order to convert the foregoing theorem into its reciprocal, we must replace the cubic U=0 by a curve of the third class, that is we must consider the coordinates which enter into the equation as line coordinates ; and it of course follows that the coordinates which enter into the equation PU = must be considered as point coordinates, that is we must consider the Pippian as a curve of the third order: we have thus the theorem; The locus of a point such that the tangents drawn from it to three given conies (the first or conic poles of any three lines with respect to a curve of the third class) form a pencil in involution, is the Pippian considered as a curve of the third order. This in fact completes the fundamental theorem in my memoirs in Liouville above referred to, and establishes the analogy with Hesse s results in relation to the Hessian ; to show this I set out the two series of theorems as follows : Hesse, in his memoirs On Curves of the Third Order and Curves of the Third Class, CreUe, tt. xxviii. xxxvi. and xxxviii. [1844, 1848, 1849], has shown as follows : (a) The locus of a point such that its polars with respect to the three conies jr = 0, F=0, Z=0 (or more generally its polars with respect to all the conies of the series \X -^ /mY •{• pZ =^0) meet in a point, is a curve of the third order F = 0. (fi) Conversely, given a curve of the third order F=0, there exists a series of conies such that the polars with respect to all the conies of any point whatever of the curve F=0, meet in a point. (7) The equation of any one of the conies in question is dx dy dz * that is, the conic is the first or conic polar of a point (X, /a, v) with respect to a certain curve of the third order 17'= 0; and this curve is determined by the condition that its Hessian is the given curve F=0, that is, we have V=HU. (S) The equation V^HU is solved by assuming U = aV+bHV, for we have then H (aV'\-bHV)=^AV + BHV, where A, B are given cubic functions of a, 6, and thence V=sHU=AV+BHV, or -4 = 1,5 = 0; the latter equation gives what is alone important, the ratio a : b; and it thus appears that there are three distinct series of conica, 400 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 each of them having the above-mentioned relation to the given curve of the third order F=0. In the memoirs in Lioumlle above referred to, I have in effect shown that — (a') The locus of a point such that the tangents from it to three conies, repre- sented in line coordinates by the equations X = 0, F=0, Z = (or more generally with respect to any three conies of the series \X + fiT-\'vZ=0) form a pencil in involution, is a curve of the third order V=0. (^) Conversely, given a curve of the third order F = 0, there exists a series of conies such that the tangents from any point whatever of the curve to any three of the conies, form a pencil in involution. Now, considering the coordinates which enter into the equation of the Pippian as point coordinates, and consequently the Pippian as a curve of the third order, I am able to add as follows : (7') The equation in line coordinates of any one of the conies in question is ^ dU dU dU ^ that is, the conic is the first or conic polar of a line (X, fi, v) with respect to a certain curve of the third class U = ] and this curve is determined by the condition that its Pippian is the given curve of the third order F = 0, that is, we have V=PU. (Sf) The equation F = Pf7' is solved by assuming U=aPV+bQVy for we have then P(aPV-\'bQV)==AV'^BHVy where A and B are given cubic functions of a, i; and thence V=^PU = AV+ BHV, or ^=1, 5 = 0; the latter equation gives what is alone important, the ratio a : b; and it thus appears that there are three distinct curves of the third class [7=0, and therefore (what indeed is shown in the Memoirs in Liouville) three distinct series of conies having the above-mentioned relation to the given curve of the third order F= 0. It is hardly necessary to remark that the preceding theorems, although precisely analogous to those of Hesse, are entirely distinct theorems, that is the two series are not connected together by any relation of reciprocity. Article Nos. 24 to 28. — VarioiLS investigations and theorems. 24. Reverting to the theorem (No. 18), that the lineo-polar envelope of the line EF is the pair of lines OE, OF; the line EF is any tangent of the Pippian, hence the theorem includes the following one: 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 401 The lineo-polar envelope with respect to the cubic, of any tangent of the Pippian, is a pair of lines. And conversely, The Pippian is the envelope of a line such that the lineo-polar envelope of the line with respect to the cubic is a pair of lines. It is I think worth while to give an independent proof. It has been shown that the equation of the lineo-polar envelope with respect to the cubic, of the line f J- -h lyy + Jf^ = (where f , i;, Jf are arbitrary quantities), is (_/.^_2f,f, -ihf-2i^. -p^-2i^, ir+^*^r. W+m if'+PM^. y. ^)'=o; and representing this equation by i (a, 6, c, /, g, h\x, y, zf = 0, we find he -/« = f (- r + 8ZV+ S^'?" + 12Pf«7?)> a6 - A« = ? (8Pf + 8ZY - ?* + 12f^^70> gh - a/= f {21' (^ + ^ + f 3) + 4; (1 + 2P) fi7?) + (1 + 8P) 7;«?«, A/-6i7 = ^(2P(r + ^+?^)+4Kl + 2P)f^?)-h(l + 8P)rr, /flr-cA = r(2/»(r + y + f') + «(l-h2?)^?)-h(l+8Z')r^^ and after all reductions, ahc-af'-hg'-ch?^2fgh or the condition in order that the conic may break up into a pair of lines is PU=0. 25. The following formulae are given in connexion with the foregoing investigation, but I have not particularly considered their geometrical signification. The lineo-polar envelope of an arbitrary line ^x-{-r)y-\'^z=0, with respect to the cubic has been represented by (a, b, c, /, g, A$a?, y, ^)' = ; and if in like manner we represent the lineo-polar envelope of the same line, with respect to a syzygetic cubic a^ + y^ + z^-{-6Vxyz=:i), by (a\ h\ c\ /, g\ hj^x, y, zf = 0, c. II. 51 402 A MEMOIR ON CUEVBS OF THE THIRD ORDER. [146 then we have a' (6c -/*) + b' (ca - g*) + c' {ab - h*) + 2f(gh- of) + 2g' {hf- hg) + 2h' (fg - ch) {l" + 2l*)(^ + r^ + ^y + (21' + 4i - S21H' + 81*) (p + ff + n f7? + (•24M'» + mH'^ - 72W' + 24P + 3) }^^, which may be verified by writing V = 1, in which case the right-hand side becomes 1 + 2P it should do, 3{PUy. If I' = «/«" » *^** ^' ^^ ^^^ syzygetic cubic be the Hessiib then the formula becomes a'{bc-f) + &c.= 361' (l + 4i» + 76Z«)(f + V + f»)» ^ + 12f' ( - 1 + 26^' + 56P) (f + .;' + f») f»7? + 121 (2 + 57^ + 168i' + 16^") fV(r* J >■ which is equal to afeK-^*^-^^'- 26. The equation {bc' + h'c-2ff,...gh'+g'h-af-a'f,...1t 17, O' = is the equation in line coordinates of a conic, the envelope of the line which cut<i-^ harmonically the conies (a, b, c, /, g, h $a:, y, zf = 0, (a', 6', c', /. gf, h'\x, t/, zy = 0; and if a, b, &c., a', &c. have the values before given to them, then the coefficients of the equation are be' + b'c-2ff = f { - f + 4«' (Z + I') iff + ^) + iUW - 21' - 21'*) fv?, ca' + e'a-2gg' =i» {-ij» + 4«' (Z + (?* + ?) + (16«'-2P- 2Z'«)fi,?, ab' + a'b- 2hh' = f ( - f»+ 4M' (Z + (? + '?*) +(16«'- 2P- 2^ ^^ gh' + g'h -af - a'f= f {(Z" + /'•) (f> + ,» + f») + {21 + 21' + 8W'>) fij?) + (1 + 4tf' (Z + O) ly^. A/' + h'f- bg' -b'g=fi {(P + 1'*) (p + ,^ + (:») + (2Z + 2^ + 8W) fijf} + (1 + 4tt' (f + f)) f»f». /5^ +/i7 - cA' - c'A = r {(P + Z'') (P + 1;* + f) + (2Z + 2Z' + 8W'«) f^f } + (1 + 4«' (f + Z')) f»,» : and we thence obtain (bc' + b'c~2/f,..,gh'+g'h-af-a'/,..Jil r,, ?)• = + ( f+ 1'*+ 16U') (? + ff+^)^^ + (6l + 6l' -l-24W»)^f» + (4 +16(W' + «'*))(,»?'+ f«f»+fS,'). =0 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 403 {is the condition which expresses that a line fa? + lyy + f^ = cuts harmonically its lineo-polar envelopes with respect to the cubic and with respect to a syzygetic cubic. 27. To find the locus of a point such that its second or line polar with respect to the cubic may be a tangent of the Pippian. Let the coordinates of the point be (a?, y, z) ; then if fa? + lyy + fer = be the equation of the polar, we have and the line in question being a tangent to the Pippian, But the preceding values give P + ^' + r' = («* + y' + '^)' + 6/(ar» + y» + ^)a?y2'-f- SGPaj^y^-^' -f ( - 2 + 8P) (y*^ + ^ar» -f ay ) fiyf = 4fP {aj" -\- y^ ■}- z^) xyz + (1 + Sl^) /c'y^z^ -^ 21 (y^z' + r^ar'-f ay); and we have therefore / (a?" + y» + -?»)' + (10i« - 1 6^) (a;* + y' + -^') a:y2: -f- (1+ 40? - 32P) a^fz'' = ; or introducing IT, HJJ in place of a:* + y* + 2:^, xyz, the equation becomes which is the equation of the locus in question. 28. The locus of a point such that its second or line polar with respect to the cubic is a tangent of the Quippian, is found in like manner by substituting the last- mentioned values of f , 17, f in the equation Q[7 = (l-10ZO(P + i;*+H-6Z«(5 + 4/*)fi75'. We find as the equation of the locus, (1 - 10i») (a;» + y» + -e»)> + 6i (1 - 30f» - 1 6/«) (a:* -h y* + ^) ay^ + 6i* (1 - 104Z' - 32/«) ji^y'z' -2(1-1- 8P)» (yV + z^a^ + a^f) = 0, where the function on the left-hand side is the octicovariant ^„U of my Third Memoir, the covariant having been in fact defined so as to satisfy the condition in question. And I have given in the memoir the following expression for 0„jr, viz. e„ir=(l-16f-6/«)f^* + (6i )U .HU + (6i' ){HUf 2(1+ 8P)» {f2^ + «»a;» + aft/'). 51—2 404 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 Article Nos. 29 to 31. — Formulod for the intersection of a ciMc curve and a line. 29. If the line ^x-^rjy -\- ^z = meet the cubic ic» + y»^ -f- 6lxyz = in the points then we have It will be convenient to represent the equation of the cubic by the abbreviate^ notation (1, 1, 1, l^x, y, 2^)^ = 0; we have the two equations (1, 1, 1, Z$^, y, ^)»=0, far-fiyy + f^ =0; and if to these we join a linear equation with arbitrary coefficients, aa -}- /3y -\- yz = 0, then the second and third equations give and substituting these values in the first equation, we obtain the resultant of the system. But this resultant will also be obtained by substituting, in the third equation, a system of simultaneous roots of the first and second equations, and equating to zero the product of the functions so obtained ^ We must have therefore (1, 1, 1, ?$^?-7i7, 7f-< a^ - /3f )' = (a^ + ^yi + 7^i) («^2 + ^ya + T-^j) (o^j + ^^3 + 7^s) ; and equating the coefficients of a', I3\ 7*, we obtain the above-mentioned relations. 30. If a tangent to the cubic a^-\- y^-\-z^ -\- 6lxyz = at a point {x^, y^, z^) of the cubic meet the cubic in the point (^,, y,, ^), then «s : ys : -2^8 = a^ (yi» - z^*) : y^ (V - x^^) : z, (x^^ - yi>). For if the equation of the tangent is f^ + lyy + f^ = 0, then and ^ : V ' S'=^i* + 2Zyi^, : yi^-\-2lziXi : z^^ + 2lxiyi. ^ This is in fact the general process of elimination given in Sohlafli's Memoir, '* Ueber die Besultante einer Systemes mehrerer algebraischer Gleichongen," Vienna Trans. 1S52. [But the prooess was employed much earlier, by Poisson.] 146] A MEMOm ON CURVES OF THE THIRD ORDER. 405 These values give = (yi' - ^i') X - (1 + 8P) x,\ since (a^, yi, ^i) is a point of the cubic; and forming in like manner the values of f* — f and f — rj\ we obtain the theorem. 31. The preceding values of (a?„ y„ z^) ought to satisfy (a^» + 2/y,2ri) a^ + (yi» -f 2h,x,) y, + (-^i* + 2lx^i) z^ = 0, ^' + yj' + ^i + 6te,ys^, = ; in fact the first equation is satisfied identically, and for the second equation we obtain x^^ + y.' + ^3' = ^a' (y 1' - ZiJ + y,' (^,' - x,J + ^,» (a:,« - y,J = - ^1' (y,' - zi') - yi' (V - ^1') - V C^i' - yiO = (^i' + yi' + ^,') (yi' - z,^) (z,' - ^j») (a:i» - y,% and consequently x/ + y»' + z,' + 6/a?,y,2r, = (a^» + y,» + ^j» + 6lx^iZ,) (y,^ - z,^) ( ^ - x,^) {x,^ - y,') = 0. which verifies the theorem. It is proper to add (the remark was made to me by Professor Sylvester) that the foregoing values ^3 : ys : ^j = a?, (yi» - z,*) : y, (z,^ - a?,') : z^ (x,^ - y,') satisfy identically the relation a?3^ + ys^ -H -^3' _ ^jM^yijff 2^ x^ysZa ^\y\Z\ Article Nos. 32 to 34. — Formulce foi* the Satellite line and point. 32. The line fa? + lyy + 5^ = meets the cubic a^^y9^2^^ Qlxyz = in three points, and the tangents to the cubic at these points meet the cubic in three points lying in a line, which has been called the Satellite line of the given line. To find the equation of the satellite line; suppose that (xi, y^, z^), (a?j, y,, z^), (a^s, y3» Z3) are the coordinates of the point in which the given line meets the cubic; then we have, as before, (1, 1, 1, «5^?-7i;, 7f-af, ai? - /3f)« = (oa^ + /9y, + 7^i)(flw?a 4^ /3y2 + 7^a)(<MJ8 + y9y, + 7-^3). 406 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 The equation of the three tangents is n = [(x,^ + 2ly,z,) X + (yi> + 2lz,x,) y + {z^^ + 2/a?iy0 z'^ =0, X [(a:,« + 2ly^;) X + (y,« + 2lz^;) y + (V + 2ir^,) 2r] and if we put F=(f» + if + ?»)'-24?»(p + i7»+f«)fi;?+(-24Z-48Z*)fV?' + (-4 + 32i0(i;'f*+?'f + fV^ (^ is the reciprocant FU of my Third Memoir), then we have identically and the equation of the satellite line is fa? + 17'^ + 5^2^ = 0. In fact the geometrical theory shows that we must have and it is then clear that iV is a mere number. To determine its value in the most simple manner, write Z = 0, y = 0, a7=f, z = — ^, we have then F, 17—^11 = 0, where The value of IT is II = F . U, and we thus obtain iV=l. For, substituting the above values, -rf(aaVai.' + &c.) + ?f»(a;,z,'z,' +&C.) - 1*^1 W, and we have and thence and consequently a^a;,«, + &c. = SJi'f, (Ci^,?, + &c. = - Sf^, x,Wz** + &c. = 9?*^ + 6?? » (y - r) = 3?«p + 6(?*i,», a;.VV + &c. = 9(7^* - 6?»f (f - '?') = 3^?* + ^^W' n= (7(17*- r)" - r(r-'?')' = (?*-?)(?+ 1?* + r - 2?''?' - Srp - 2pi;'). 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 407 Now considering the equation in order to find f, ri\ f it will be sufficient to find the coefficients of a:*, y", z^ in the function on the left-hand side of the equation. The coefficient of a^ in 11 is {a;,^ + 2lyiZi) (irj" + 2lz^^) (a^* + 2Za?,ys) -^ •Crj •C/2 «*^ + 2? {onxx^y^z + &c.) + 4i'» (x^y^^^t + &c.) • +8Z» yiy^i;siz^t\ and it is easy to see that representing the function (1, 1, 1, /$)8?- 717, 7f - a?, ai7 - fi^f by (a, b, c, f, g, h, i, j, k, l$a, /8, 7)», the symmetrical functions can be expressed in terms of the quantities a, b, &c., and that the preceding value of the coefficient of a^ in H is a' -f 21 (9hj - 6al) + 4Z» (6gk - 3fj - Shi + 31») + 8Z' be; and substituting for a, &c. their values, this becomes + 4^' {- 6 (?r + 21^) (pi, + 2Zf C) + 3(i,{:» + 2f?p)((7f + 2i5i;«) + 3 (?.;' + 2?i,r') (»?•?+ 2^P)) and reducing, we obtain for the coefiScient of a* in 11 the following expression, -18/ ^*^ - 24? (p + 1;" + {:») fi^r -24Z»(i7»i;»+C»|» + f.7») 408 A M£MOIR ON CURVES OF THE THIRD ORDER. [146 Now the coefficient of a^ in F , U is simply F, which is equal to f«+ ^«+ f»- 2i;»?»- 2f*f - 2f i7» and subtracting, the coefficient ofa^ in F. U —Tl is f - 2f i;» - 2f ?» - 48^f^^?», which is equal to (1 + 8P) f (f* - 2fi;» - 2f ?» - 6lvV)' The expression last written down is therefore the value of pf, or dividing by ^ we have f, and then the values of rj', f are of course known, and we obtain the identical equation F. U^U = (l + 8P)(f^+W+?^)' ^ + (17* + (?* 2^7» - 2^f» 2i7?»-2i;p 2rr - 2?i;» 6zr-r)y and the second factor equated to zero is the equation of the satellite line of fa? + i;y + fg: = 0. 33. The point of intersection of the line f a? + lyy + f-^ = with the satellite line ^'x + i;'y + f -^ = is the satellite point of the former line ; and the coordinates of the satellite point are at once found to be : (P-V)(f^ + 2in- 34. If the primary line ^x + rfy + fy = is a tangent to the cubic, then (a?,, y,, ^i) being the coordinates of the point of contact, we have ^ : 1) ■ ?=«i' + 2/yi^, : yi*-\r'ilz^ : «i*+2^,; 146] A MEMOIR ON CURVES OF THE THIRD ORDER. 409 these values give as before and they give also and consequently we obtain that is, the satellite point of a tangent of the cubic is the point in which this tangent again meets the cubic. Article Nos. 35 and 36. — Theorems relating to the satellite point. 35. If the line f a? 4- i;y + ?2r = be a tangent of the Pippian, then the locus of the satellite point is the Hessian. Take (x, y, z) as the coordinates of Jbhe satellite point, then we have x:y : ^ = (i;» - ?») (17?+ 2/p) :(C-P)(Sf+2Zi;») where the parameters f, 1;, 1^ are connected by the equation -i(?+i;'+C')+(-i+4P)fi;r=0. We have &ad it is easy to see that the function on the right-hand side must divide by tf — i^: hence a^ + y' + s^ will also divide by 17* — f*, and consequently by (ij* - {?) (S* — ?) (? — «?*)• We have I + ? + 6?fV?* { - ^ - rV - 1/' + 3p (?» + 1;») - 3f ) + 12?^,? {-rf^(rf+^) + 3p,^C - f } + 8i» { - i/'f + 3i;»r? - (V + D PI and C. II. 52 410 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 Adding these values and completing the reduction, we find (*■» + y* + z») ^ (»?'- f) (?'-?) (f» - »7») = - ^ - 1?« -?•+ Zi/T + 2{:»? + 2f»7' + 18i ^*^ + 12P(|' + i7' + f»)fi,? + 8i» (i/'C + Cf + fi?») ; and we have also + 8Z' f«v'r, and thence -A i^ + ri' + ^y + (UPA + ilB) (f* + 17» + f») fi^r + (18^4 + (1 + 8?) 5) pi,«i;» + ((4P + 81*) A + 4P5) (vT + rf + f 'j')- The coefficient of ij«?» + f »f» + f <;» on the right-hand side will vanish if (1 + 2?) -4 + i* 5 = 0, or, what is the same thing, if A=l^, 5 = — (1 + 2Z'); and substituting these values, we obtain {P {x> + y' + z>)-(l + 2P) f,C} H- (i;* - ?») (C - p) (p - ,;») = -P (f + 'T' + H + (- 4Z + 4i*) (t' + 1?' + f) fijf + (- 1 + 8P - lei*) f*ij»?», or, what is the same thing, P (a;* + y" + ^) - (1 + 2P) a;y^ = - (i,» - H (r - f ) (? -• 7*) x[-li^ + v'+^) + (-l + U*) f.7?}'. Hence the left-hand side vanishes in virtue of the relation between ^, rj, ^, or we have ^{x> + f-i-z')-{l + 2P)xyz = 0. which proves the theorem. 36. Suppose that {X, Y, Z) are the coordinates of a point of the Hessian, and let (P, Q, R) be the coordinates of the point in which the tangent to the Hessian at the point {X, Y, Z) again meets the Hessian, or, what is the same thing, the 146] A MEMOIB ON CURVES OF THE THIRD ORDER. 411 satellite point in regard to the Hessian of the tangent at (X, F, Z), And consider the conic X{x'^' 2lyz) + Tif-^^ 2lzx) + Z(a^'{- 2lxy\ which is the first or conic polar of the point (X, F, Z) in respect of the cubic. The polar (in respect to this conic) of the point (P, Q, R) will be w here f = PZ + /(i2F+(2Z), i=RZ + l(QX+PY); or putting for (P, Q, iJ) their values, ^ = (Y''-Z')(X*-IYZ), V = (Z'-X')(7^^1ZX), r = (Z»-P)(^-iZF); and if from these equations and the equation of the Hessian we eliminate {X, F, Z) we shall obtain the equation in line coordinates of the curve which is the envelope of the line fa? + i;y + 5^ = 0. We find, in fact, p + ,7»+ r = (F« - iP'XZ'- Z») (Z» - F«) SI {X'+T' + Z*)XYZ + 91* X^Y^Z^ (,+ (!- W){Y^Z* + ^Z» + Z^F'), X -^ ^? = (F'-^)(^ - X»)(X»- P) r J Z'(-y*+F' + ^)ZF^ i (7'^ + ^Z» + Z»F'); and thence recollecting that HU^J? (Z» + F» + -Z») - (1 + 2P) ZF-^, we find and the equation of the envelope is which is therefore the Pippian. We have thus the theorem: 52—2 412 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 The envelope of the polar of the satellite point in respect to the Heesian of the tangent at any point of the Hessian, such polar being in respect of the conic which is the first or conic polar of the point of the Hessian in respect of the cubic, is the Pippian. Article Nos. 37 to 40. — Investigations and theorems relating to the first or conic polar of a point of the cubic. 37. The investigations next following depend on the identical equations [a(X^ + 2lYZ)-{-l3{Y'-^2lZX)-^y(Z^ + 2lXY)] = {Z (iB* + 2lyz) + F (y« + 2lzx) + Z (-«» + 2%)} -h {x (Z« + 21YZ) + y (F» + 21ZX) '{-z{Z'-\- 21XY)\ x{-(aF^-fi9ZZ-f7ZF)(Za;»+Fy» + Z^«) + (aZ« + i9F» + 7Z*)(ZF?+Fzx + Za;y)l. which is easily verified. I represent the equation in question by then considering (a?, y, z) as current coordinates, and (Z, Y, Z) and (a, /8, 7) as the coordinates of two given points 2 and fl, we shall have 17=0 the equation of the cubic, TT = the equation of the first or conic polar of S with respect to the cubic, P = the equation of the second or line polar of 2 with respect to the cubic The equation T = is that of a syzygetic cubic passing through the point 2 : the coordinates of the satellite point in respect to this syzygetic cubic of its tangent at 2 are XiY^'-Z') : Y{Z^^X^) : Z(Z»- F»); and calling the point in question 2^ then Z = is the equation of a line through the points 2', ft. The equation = is that of a conic, viz. the first or conic polar of 2 with respect to a certain sjrzygetic cubic - 2 {aYZ-^^ZX + 7ZF) («» + y» + ^) + (aZ» + /8F» + 7^)ajyir = 0, depending on the points 2, ft, or, what is the same thing, the conic B = is a properly selected conic passing through the points of intersection of the first or conic polars of 2 with respect to any two sjrzygetic cubics; and lastly, .E' is a constiint coefficient. The equation expresses that the points of intersection of (ir=o, p = o), (Tr=o, e = o), (Z = o, p=o), (Z = o, e = o), lie in the syzygetic cubic T = 0. 146] A MEMOIB ON CURVES OF THE THIRD OKDEB. 413 The left-hand side of the equation may be written - ZF^ {a (Z« + 2ZFZ) + /8 ( F« + 2/ZZ) + 7 (^ + 2iZF)l(rc» + y« + ^ + eZfljy^) + xyz{a(X^ + 2l7Z)+fi(7^ + 2lZX) + y{Z» + 2lXY)}(X^-^Y^ + Z' + 6lXYZ); and it may be remarked also that we have -3ZFZ{a(Z» + 2iFZ) + /9(F« + 2ZZZ) + 7(^ + 2/ZF)} equal identically to (Z(F»-Z»)(7F-/8^)+F(Z*-.Z»)(a^-7Z) + ^(Z«-F»)(i8Z-aF)} -(aFZ + /SZZ + 7ZF)(Z«+ Y' -^ Z' + eiXYZ). Hence if we assume Z»+ F» + Z» + 6?ZFZ=0, the equation will take the form where the constant coefficient K may be expressed under the two different forms if = -ZFZ{a(Z« + 2iFZ) + /8(F» + 2/ZZ) + 7(^ + 2ZZF)} = J {Z(F' - Z*) (7F- i8Z)+ F(^- Z«)(aZ- 7Z) + Z(Z»- Y*){^X - aF)}, and W, Z, P, have the same values as before. In the present case the point S is a point of the cubic : the equation IT = represents the first or conic polar of the point in question, and the equation P = its second or line polar, which is also the tangent of the cubic. The line Z = is a line joining the point XI with the satellite point of the tangent at S, or dropping altogether the consideration of the point ft, is an arbitrary line through the satellite point: the first or conic polar of 2 meets the cubic twice in the point 2, and therefore also meets it in four other points; the conic © = is a conic passing through these four points, and com- pletely determined when the particular position of the line through the satellite point is given. And, as before remarked, = is a conic passing through the points of intersection of the first or conic polars of 2 with respect to any two syzygetic cubics. We have thus the theorem : The first or conic polar of a point of the cubic touches the cubic at this point, and besides meets it in four other points; the four points in question are the points in which the first or conic polar of the given point in respect of the cubic is iutersected by the first or conic polar of the same point in respect to any syzygetic cubic whatever. 38. The analytical result may be thus stated: putting K = ^YZ'^/3ZX + yXY, X = aZ« + /SF» + 7Z«, 414 A MEMOm ON CUBVES OF THE THULD ORDER. [146 or, if we please, considering k, \ as arbitrary parameters, then the four points lie in the conic (2kX, 2/cT, 2/cZ, -\Z, -\F, -XZ$a?, y, 2r)> = 0, or, what is the same thing, they are the points of intersection of the two conies Za^ + Fy« + Zz^ = 0, Xyz + Yzx + Zxy = 0. 39. Considering the four points as the angles of a quadrangle, it may be shown that the three centres of the quadrangle lie on the cubic. To effect this, assume that the conic (2/cZ, 2k7, 2kZ, -XZ, -\F, -\Z\x, y, z>^=^0 represents a pair of lines; these lines will intersect in a point, which is one of the three centres in question. And taking x, y, z a& the coordinates of this point, we have a^ : y^ \ z^ : yz I zx I xy ^ 4/^* YZ — X'Z* 4/e»ZZ- X»F» 4/e» XT" X»^ X» YZ + 2k\X^ X»ZZ + 2/eXF« X«ZF+2/eX^; and we may, if we please, use these equations to find the relation between /c, X Thus in the identical equation a^ . y* — (fl?y)" = 0, substituting for a^, xy, y" their values, and throwing out the factor Z, we find (4#c>-X»)ZFZ-/eX«(Z«+ F« + ^) = 0, and thence, in virtue of the equation Z'+ F* + ^ + 6?ZFZ = 0, we obtain 4#c> -f- 6?/eX» - X» = 0. But the preceding system gives conversely, X* : Y* : Z" : YZ : ZX : XY^^i^z- XV ^i^zx - xy 4/c'a?y — XV Xh/z + 2/cXflj« X^zx + 2/cXy* X^xy + 2/eX«». Hence firom the identical relation Z*. F»-(ZF)» = 0, substituting for X\ XY, P their values, and throwing out the £sbctor z, we find (4ie* — X')a5y2r — /eX*(«* + y' + z*) = 0, and thence, in virtue of the equation 4#c* — X' = — 6i/icX*, we obtain ^ + y* + ^ + Qlxyz = 0, 146] A MEMOIB ON CURVES OF THE THIRD ORDER. 415 which shows that the point in question lies on the cubic. We have thus the theorem : The first or conic polar of a point of the cubic touches the cubic at the point, and meets it besides in four points, which are the angles of a quadrangle the centres of which lie on the cubic. In other words, the quadrangle is an inscribed quadrangle. 40. To find the equations of the three axes of the quadrangle, that is of the lines through two centres. We have (4/e» 7Z - \^X') x + ( \'XY-^ ^KkZ") y + ( X»^Z + 2/eX7«) ^ = 0, ( \^XY-{- 2k\Z^) X + (4^ZX - X«r») y + ( X'FZ + 2ic\Z«) ^ = 0, ( X'ZZ + 2/rXF>) a? + ( X«7Z + 2/rXZ»)y+(4/e»ZF- X«Z0-r=O; or arranging these equations in the proper form and eliminating ic", k\ X', we find YZx, Z^-\-Y^z, X{-Xx-^Yy-\-Zz) =0; ZXy, XH^-Z'x , Y{ Xx-Yy-^- Zz) XYz, r»a? + Z«y, Z{ Xx -\- Yy -\- Zz) or, multiplying out, ZFZ{(^^F»)a;» + (Z»-^)y» + (r»-Z»)-»'} -f a^yZY^ (- 2Z» + F» + ^) + zai'YZ^ (2X^ --Y^^Z*) + y'zXZ^ (- 2 F» + ^ -f Z») + xfZX* (2F» - 2> - Z») + z'xYX*(- 2Z* + Z» + F») + y^»ZF'(2^ - Z* - F») = 0. We may simplify this result by means of the equation Z»+ F» + ^ + 6ZZFZ= 0, so as to make the left-hand side divide out by XYZ: we thus obtain (^-F»)a:» + (Z»-i?')y» + (F»-Z»)-?' + (- 3Z»F-. 6iF>Z)^ + (- 3F»Z- 6lZ'X)y*z + (- 3^Z - 6lX'Y)^x + ( 3ZF«+6ZZ>Z)«y« + ( 3FZ*+6/F»Z)y^»-|-( 3^Z^ + 6/^F)^a^ = (); or in a difierent form, + (- Sx'y - eiz^x) X^Y+(- 3y'z - Gla^^y) Y'Z + (- 3z'x - 6ly^z) Z'X + ( fixy''h6lyz^)XY^ + ( Syz^ + 6lzaf) YZ' + ( Szx' -\- Qlxf) ZX^ == 0, as the equation of the three axes of the quadrangle. 416 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 Article No. 41. RecapitulcUum of geometrical definxtione of the Pippian. In conclusion, I will recapitulate the different modes of generation or geometrical definitions of the Pippian, obtained in the course of the present memoir. The curve in question is: 1. The envelope of the line joining a pair of conjugate poles of the cubic (see Nos. 2 and 13). 2. The envelope of each line of the pair forming the first or conic polar with respect to the cubic of a conjugate pole of the cubic (see Nos. 2 and 14). 3. The envelope of a line which is the polar of a conjugate pole of the cubic, with respect to the conic which is the first or conic polar of the other conjugate pole in respect to any syzygetic cubic (see Noa 2 and 9). 4. The locus of the harmonic with respect to a pair of conjugate poles of the cubic of the third point of intersection with the Hessian of the line joining the two conjugate poles (see Nos. 2 and 17). 6. The envelope of a line such that its lineo-polar envelope with respect to the cubic breaks up into a pair of lines (see No. 24). 6. The envelope of a line which meets three conies, the first or conic polars of any three points in respect to the cubic, in six points in involution (see No. 22^ 7. The envelope of the second or line polar with respect to the cubic, of a point the locus of which is a certain curve of the sixth order in quadratic syzygy with the cubic and Hessian, viz. the curve — S . IT"* + (fiVy = (see No. 27). 8. The envelope of a line having for its satellite point a point of the Hessian (see No. 35). 9. The envelope of the polar of the satellite point with respect to the Hessian of the tangent at a point of the Hessian, with respect to the first or conic polar of the point of the Hessian in respect to the cubic (see Na 86). 147. I I A MEMOIR ON THE SYMMETRIC FUNCTIONS OF THE ROOTS OF AN EQUATION. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 489 — 499. Received December 18, 1856, — Read January 8, 1857.] There are contained in a work, which is not, I think, so generally known as it deserves to be, the "Algebra" of Meyer Hirsch [the work referred to is entitled Sammlung von Beispielen Formeln und Aufgaben aus der Buchstabenrechnung und Algebra, 8vo. Berlin, 1804 (8 ed. 1853), English translation by Ross, 8vo. London, 1827] some very useful tables of the symmetric functions up to the tenth degree of the roots of an equation of any order. It seems desirable to join to these a set of tables, giving reciprocally the expressions of the powers and products of the coefficients in terms of the symmetric functions of the roots. The present memoir contains the two sets of tables, viz. the new tables distinguished by the letter (a), and the tables of Meyer Hirsch distinguished by the letter (6) ; the memoir contains also some remarks as to the mode of calculation of the new tables, and also as to a peculiar symmetiy of the numbers in the tables of each set, a synmietry which, so far as I am aware, has not hitherto been observed, and the existence of which appears to constitute an important theorem in the subject. The theorem in question might, I think, be deduced from a very elegant formula of M. Borchardt (referred to in the sequel), which gives the generating function of any sjonmetric function of the roots, and contains potentially a method for the calculation of the Tables (6), but which, fix)m the example I have given, would not appear to be a very convenient one for actual calculation. Suppose in general (1, 6, C...5I, a?)* =(1 — 6a?)(l— )8a:)(l — 7a:)... , so that — 6 = 2a, + c = 2a)8, — d = %afiy, &c., and if in general 0. n. 53 418 A MEMOIB ON THE SYMMETRIC FUNCTIONS [147 where as usual the summation extends only to the distinct terms, so that e.g. (/?*) contains only half as many terms as (pq), and so in all similar cases, then we have -6 = (l), +c = (P), -d = (P), &c.; and the two problems which arise are, first to express any combination fr'^c^... in terms of the symmetric functions (l^m^..,), and secondly, or conversely, to express any symmetric function {l^m^ ...) in terms of the combinations b^d^,... It will conduce materially to brevity if 1^29... be termed the partition belonging to the combination b^d^... ; and in like manner if l^m^... be termed the partition belonging to the symmetric function (l*m^,,,), and if the sum of the component numbers of the partition is termed the weight. Consider now a line of combinations corresponding to a given weight, e.g. the weight 4, this will be e bd d" b^c b" (line) 4 13 2» 1«2 1*, where I have written under each combination the partition which belongs to it, and in like manner a colunm of symmetric functions of the same weight, viz. (4) (column) (31) (2») (21*) (1*). where, as the partitions are obtained by simply omitting the ( ), I have not separately written down the partitions. It is at once obvious that the different combinations of the line will be made up of numerical multiples of the symmetric functions of the column ; and conversely, that the symmetric functions of the column will be made up of numerical multiples of the combinations of ^ the line ; but this requires a further examination. There are certain restrictions as to the symmetric functions which enter into the expression of the com- bination, and conversely, as to the combinations which enter into the expression of the symmetric function. The nature of the first restriction is most clearly seen by the following Table: Number of Greatest Parts. Part. 1 4 2 3 2 2 3 2 4 1 Combinations with their several Partitions. 6 bd b^o b* 4 13 2« 1«2 1* Contain Maltiples of the Symmetrio Funotions. (1*), (21»), (21'), (2'), (2^), (31), (2'), (31), (4) Greatest Part does not exceed 1 2 2 3 4 Number of j Parts not less than 4 3 2 2 1 147] OF THE ROOTS OF AN EQUATION. 419 Thus, for instance, the combination bd (the partition whereof is 13) contains multiples of the two symmetric functions (1*), (21*) only. The number of parts in the partition 13 is 2, and the greatest part is 3. And in the partitions (1*), (21') the greatest part is 2, and the number of parts is not less than 3. The reason is obvious : each term of the developed expression of bd must contain at least as many roots as are contained in each term of d, that is 3 roots, and since the coefficients are linear functions in respect to each root, the combination bd cannot contain a power higher than 2 of any root. The reasoning is immediately applied to any other case, and we obtain First Restriction. — A combination b^c^... contains only those symmetric functions (l*mv.,,), for which the greatest part does not exceed the number of parts in the partition P2^... , and the number of parts is not less than the greatest part in the same partition. Consider a partition such as 1*2, then replacing each number by a line of units thus, 1 1 11, and summing the columns, we obtain a new partition 31, which may be called the conjugate^ of 1*2. It is easy to see that the expression for the combination 6*c (for which the partition is 1'2) contains with the coefficient unity, the symmetric function (31). the partition whereof is the conjugate of 1*2. In fact 6"c = (— 2a)" (Sa/8), which obviously contains the term + lo*^, and therefore the symmetric function with its coefficient + 1 (31) ; and the reasoning is general, or Theorem. A combination 6^c^... contains the sjrmmetric function (partition conjugate to 1''2^...) with the coefficient unity, and sign + or — according as the weight is even or odd. Imagine the partitions arranged as in the preceding column, viz. first the partition into one part, then the partitions into two parts, then the partitions into three parts, and so on; the partitions into the same number of parts being arranged according to the magnitude of the greatest part (the greatest magnitude first), and in case of equality according to the magnitudes of the next greatest part, and so on (for other examples, see the outside column of any one of the Tables). The order being thus completely defined, we may speak of a partition as being prior or posterior to another. We are now able to state a- second restriction as follows. Second Restriction. — The combination b^cfi.,. contains only those symmetric functions which are of the form (partition not prior to the conjugate of 1*2*...). The terms excluded by the two restrictions are many of them the same, and it might at first sight appear as if the two restrictions were identical; but this is not ^ The notion of Conjugate Partitions is, I believe, due to Professor Sylvester or Bfr Ferrers. [It was dne to Mr now Dr Ferrers.] 53—2 420 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 so : for instance, for the combination bd\ see Table VII (a), the term (41*) is excluded by the first restriction, but not by the second; and on the other hand, the term (3*1), which is not excluded by the first restriction, is excluded by the second restriction, as containing a partition 3^1 prior in order to 32', which is the partition conjugate to 13', the partition of bd\ It is easy to see why b(P does not contain the symmetric function (3'1); in fact, a term of (3*1) is (a'/3'7), which is obviously not a term of 6d' = (— 2a) (Sa^y)* ; but I have not investigated the general proof I proceed to explain the construction of the Tables (a). The outside column contains the symmetric fimctions arranged in the order before explained; the outside or top line contains the combinations of the same weight arranged as follows, viz. the partitions taken in order from right to left are respectively conjugate to the partitions in the outside column, taken in order from top to bottom ; ia other words, each square of the sinister diagonal corresponds to two partitions which are conjugate to each other. It is to be noticed that the combinations taken in order, from left to right, are not in the order in which they would be obtained by Arbogast's Method of Derivations from an operand a*, a being ultimately replaced by unity. The squares above the sinister diagonal are empty (i.e. the coefficients are zero), the greater part of them in virtue of both restrictions, and the remainder in virtue of the second restriction; the empty squares below the sinister diagonal are empty in virtue of the second restriction; but the property was not assumed in the calculation. The greater part of the numbers in the Tables (a) were calculated, those of each table fi"om the numbers in the next preceding table by the following method, depending on the derivation of the expression for b^^^c^... from the expression for b'^c^... Suppose, for example, the column cd of Table V(a) is known, and we wish to calculate the column bed of Table VI (a). The process is as follows : Given we obtain 2n 21V 1» 3 10 321 2' 31» 2n« 21* 1« 1 3 2 3 6 12 10 60 1 3 3 8 22 60 where the numbers in the last line are the numbers in the column bed of Table VI (a). The partition 2*1, considered as containing a part zero, gives, when the parts are successively increased by 1, the partitions 321, 2', 2*1', in which the indices of the increased part (i.e. the original part plus unity) are 1, 3, 2; these numbers are taken as multipliers of the coefficient I of the partition 2'1, and we thus have the new coefficients 1, 3, 2 of the partitions 321, 2*, 2*1*. In like manner the coefficient 3 of 147] OF THE ROOTS OF AN EQUATION. 421 the partition 21' gives the new coefficients 3, 6, 12 of the partitions 31', 2*1*, 21*, and the coefficient 10 of the partition 1* gives the new coefficients 10, 60 of the partitions 21* and 1", and finally, the last line is obtained by addition. The process in fact amounts to the multiplication separately of each term of cd = 1 (2»1) + 3 (21') + 10 (1») by 6 = (1). It would perhaps have been proper to employ an analogous rule for the calculation of the combinations M^.,. not containing 6, but instead of doing so I availed myself of the existing Tables (6). But the comparison of the last line of each Table (a) (which as corresponding to a combination b^ was always calculated in- dependently of the Tables (6)) with such last line as calculated from the corresponding Table (6), seems to afford a complete verification of both the Tables ; and my process has in fact enabled me to detect several numerical errors in the Tables (6), as given in the English translation of the work above referred to. It is not desirable, as regards facility of calculation and independently of the want of verification, to calculate either set of Tables wholly from the other; the rules for the independent calculation of the Tables (6) are fully and clearly explained in the work referred to, and I have nothing to add upon this subject. The relation of symmetry, alluded to in the introductory paragraph of the present memoir, exists in each Table of either set, and is as follows: viz. the number in the Table corresponding to any two partitions in the outside column and the outside line respectively, is equal to the number corresponding to the same two partitions in the outside line and the outside column respectively. Or, calling the two partitions P, Q, and writing for shortness, combination (P) for the combination represented by the partition P, and for greater clearness, symmetric function (P) (instead of merely (P)) to denote the symmetric function represented by the partition P, we have the following two theorems, viz. Theorem. The coefficient in combination (P) of symmetric function (Q) is equal to the coefficient in combination (Q) of symmetric function (P); and conversely. Theorem. The coefficient in symmetric function (P) of combination (Q) is equal to the coefficient in symmetric function (Q) of combination (P). M. Borchardt's formula, before referred to, is given in the *Monatsbericht' of the Berlin Academy (March 5, 1885) S and may be thus stated; viz. considering the case of n roots, write (1, 6, c, ... k"$l, xf = (1 - aa;)(l - ^a?)...(l - kx) =/b, then 1 1 _i ^N ^ 1 / x„ My-'-f^ A ^ A. n(^,y,...ti) J -ax l--^y'"l- Ku) k^ ^ Il(Xyy,...u) dx dy" du fxjy...fu ' 1 And in Crelle, t. liii. p. 195.— Note added 4th Deo. 1857, A. C. 422 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 where 11 (x, y,.,,u) denotes the product of the differences of the quantities x, y,..,v, and on the left-hand side the summation extends to all the different permutations of a, ^, ... /t, or what is the same thing, of x, y,...u. Suppose for a moment that there are only two roots, so that (1, 6, cth xy^(l^ax)(l^l3xl then the left-hand side is 1 1 which is equal to 2 + (a + i9)(a? + y) + (a» + ^)(a;» + y') + 2a/8ajy + (a» + /80(aJ» + y') + (a»i8 + <^^ and the right-hand side is 1 fx/y d d x-y which is equal to c ' x-y dx dy fxfy ' 1 My [ r^fy-ryfaH^-y)rxfy \ c x — y (f^r {fyf and therefore to c fxfy { x-y J J i^y or substituting for fx^ fy their values, f^fy -fyfa x-y becomes equal to 2c - 62 - 6c(a; + y) - 2c»icy, and fxfy is equal to 6» + 2ic (a? H- y) + ^xy. The right-hand side is therefore equal to 2 + 6(a? + y) + 2cay . (l+6a? + cic»)(l + 6y + cy«)' and comparing with the value of the left-hand side, we see that this expression may be considered as the generating function of the sjrmmetric functions of (a, /S), viz. the expression in question is developable in a series of the symmetric functions of (a:, y\ the coefficients being of course functions of 6 and c, and these coefficients are (to given numerical factors yrls) the symmetric functions of the roots (a, /8). 147] OP THE ROOTS OP AN EQUATION. 423 And in general it is easy to see that the left-hand side of M. Borchardt's formula is equal to [0] + [1] (1) (ly + [2] (2) (2)' + [P] (1») (1')' + &c, where (1), (2), (1'), &c. are the symmetric functions of the roots (a, A ••• fc)y (1)', (2)', {V)\ &c. are the corresponding symmetric functions of («, y, ...w), and [0], [1], [2], [1*], &c. are mere numerical coefficients; viz. [0] is equal to 1.2.3...n, and [1], [2], [1'], &c. are such that the product of one of these factors into the number of terms in the corresponding sjrmmetric function (1), (2), (P), &c. may be equal to 1.2.3...n. The right-hand side of M. Borchardt's formula is therefore, as in the particular case, the generating function of the symmetric functions of the roots (a, ^, ... ^), and if a convenient expression of such right-hand side could be obtained, we might by means of it express all the sjrmmetric functions of the roots in terms of the coefficienta Tahles rdating to the Symmetric Functions of the Roots of an EquaMon. The outside line of letters contains the combinations (powers and products) of the coefficients, the coefficients being all with the positive sign, and the coefficient of the highest power being unity; thus in the case of a cubic equation the equation is a;^ + 6«* + w? + d = (a? - a) (a? - )8) (a? - 7) = 0. The outside line of numbers is obtained from that of letters merely by writing 1, 2, 3... for 6, c, d..., and may be considered simply as a different notation for the combinations. The outside column contains the different symmetric functions in the notation above explained, viz. (1) denotes So, (2) denotes So", (1') denotes SayS, and so on. The Tables (a) are to be read according to the columns; thus Table 11(a) means 6" = 1 (2) + 2(1)*, c = (l*). The Tables (6) are to be read according to the lines; thus Table 11(6) means (2) = - 2c + 1ft*, (1») = + Ic. 1(a). (1)1-1 1(6). (1) 1 6 - 1 II (a). (2) (I'O 2 c + 1 1» + 1 + 2 II (6). 2 c P 6* (2) (1') m(o). 3 12 1» 11 d he 6» (3) - 1 (21) - 1 -3 (1') -1 -3 -6 m (6). 3 12 1» d he 6» (3) -31 + 3 - 1 (21) + 3-1 (1») - 1 424 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 IV (a). (5) (41) (32) (31«) (2'1) (2P) (!•) II (4) (31) (2«) (21') (1^) 4 13 2^ P2 V e bd c* b^c b* + 1 + 1 + 1 + 2 + 4 + 6 + 1 + 1 + 4 + 2 + 6 + 5 + 12 + 12 + 24 V(o). 5 / - i 14 be - 1 -5 23 cd P3 b*d 12* 6c« P2 b^c b' 1 - 3 1 - 5 1 - 2 - 10 - 1 - 7 - 12 -27 - 20 - 1 - 2 - 7 - 5 - 30 - 60 - 3 10 -12 -20 -30 -60!- 120 IV (b). i 13 2« 1«2 I* ^ « bd c« ft»c 6* (4) -4 + 4 + 2 - 4 + 1 (31) + 4 - 1 -2 + 1 (2') + 2 -2 + 1 (21') -4 + 1 (1') + 1 V (6). 5 14 23 P3 12« P2 P = / be cd b^d b<^ 6»c 6» (5) -5 + 5 + 5 -5 5 + 5 -1 (41) + 5 - 1 -5 + 1 + 3 - 1 (32) + 5 -5 + 1 + 2 - 1 (31») -5 + 1 + 2 - 1 (2*1) - 5 + 3 - 1 (2P) + 5 - 1 (P) - 1 1 II 6 9 15 ¥ + I 24 ce P4 b^e 3» d" VI (a] 123 bed 1. P3 b^d 2» P2« P2 b'c 6« (6) i + 1 (51) + 1 + 6 (42) + l!+ 4 + 15 (3') + 1 + 2 + 6 + 20 (41') + 1 • • • + 2 + 9 + 30 (321) + 1 + 3 + 3 + 6 + 8 + 22 + 60 (2') + 1 + 3 + 6 + 15 + 36 + 90 (31') + 1 • • • + 3 + 10 + 6 + 18 + 48 + 120 (2»1') + 1 + 2 + 2 + 8 + 18 + 15 + 34 + 78 + 180 (21*) + 4 + 15 + 9 + 6 + 22 + 48 + 36 + 78 + 168 + 360 (1-) + 1 + 6 + 30 + 20 + 60 + 120 + 90 + 180 + 360 + 720 VI (6). (6 (51 (42 (3« (4P (321 (2" (31 (2«P (2P 6 9 15 ¥ + 6 24 ee + 6 P4 6»c -6 + 1 + 2 + 3 - 1 -3 • • • + 1 3« d* + 3 -3 -3 + 3 + 3 -3 + 1 123 bed P3 b^d + 6 - 1 - 2 • • • + 1 2» c» - 2 + 2 2 + 1 P2> 6V + 9 -4 + 1 P2 b*c -6 + 1 1* + 1 - 6 - 12 + 6 -l|-6 + 7 + 6 -6 -3 + 1 + 7 + 2 - 1 -4 + 1 + 2 -3 + 2 + 4 -2 2 + 1 + 4 + 3 - 3 - 6 - 3 - 12 + 1 - 2 + 6 + 9 - 6 + 1 147] OP THE ROOTS OF AN EQUATION. 425 7 h 16 ^9 25 1«5 34 de 124 hce 1»4 6»« VII ( 12« [a), 2«3 c^d P23 h^cd P3 h^d 12» 6c» P2« 6V P2 6»c V - 1 -7 1 1 7 - 21 1 5 - 1 3 - 10 - 35 - 42 - 1 • • • 2 - 11 1 - 1 - 4 - 3 - 11 - 35 - 105 - 1 - 2 - 6 - 7 - 18 - 50 - 140 - 1 - 2 - 5 - 12 - 12 - 31 - 80 - 210 1- 1 • • • • • • - 3 - 13 - 6 - 24 - 75 - 210 - 1 - 3 - 2 - 5 - 13 - 34 - 27 - 68 - 170 - 420 - 1 - 3 - 6 - 13 - 7 - 12 - 27 - 60 - 51 - 117 270 - 630 - 1 • • • - 4 - 6 12 - 34 - 88 - 60 - 150 - 360 - 840 - 1 - 2 - 3 - 11 - 24 - 18 - 31 - 68 - 150 -117 - 25« - 570 - 1260 - 5 - 11 -10 -35 - 35 - 75 - 50 - 80 - 170 -360 -270 - 570 - 1200 -2520 - 1 -21 -42 - 105 -210 - 140 -210 -420 -840 -630 - 1260 -2520 -5040 (7) (61) (52) 7 h - 7 + 7 + 7 16 hg + 7 - 1 -7 -7 + 1 + 8 + 4 + 7 - 1 -9 -5 + 1 + 5 - 1 25 cf + 7 -7 + 3 -7 + 2 + 4 + 7 -3 - 2 -6 + 3 + 2 - 1 1»5 bV -7 + 1 + 2 + 7 - 1 -3 -4 -2 + 1 + 4 • • • - 1 34 de + 7 -7 -7 + 5 + 7 + 2 -5 + 1 -3 + 3 - 1 124 bee - 14 + 8 + 4 + 2 - 3 - 8 + 1 + 2 + 3 - 1 VII 1'4 6»e + 7 - 1 -2 -3 + 1 + 3 • • • • • • - 1 (6). ll« bd^ -7 + 4 + 7 -5 -4 + 1 + 2 - 1 2«3 cH -7 + 7 -3 + 1 -2 + 2 - 1 1 P23 h'cd + 21 - 9 - 6 + 3 + 4 - 1 P3 b*d -7 + 1 + 2 • • • - 1 12' b(? + 7 -5 + 3 - 1 1322 - 14 + 5 - 1 1»2 + 7 - 1 V b' - 1 (43) (5P) + 7 - 7 - 14 - 7 (421) (3*1) 1 (32«) (4P) (321«) (2*1) (31^) (2n») (2P) (10 - 7 + 7 + 21 + 7 - 7 - 14 + 7 - 1 1 ■ c. n. 54 426 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 VIII (a). Runs on infrit. (8) (71) (62) (53) (4') (6P) (521) (431) (42») (3''2) (51') (421«) (3'1») (32n) (2*) (41') (321') (2'1»; (31»; (2'1«) (21') (P) 8 • 1 17 bh 26 eg P6 35 df 125 be/ V5 4^ 134 bde 2«4 1«24 b'^ce P4 ; 23« b*e \ cd* 1 1 1 1 : 1 1 • — 1 1 1 1 1 i 1 1 — — 1 1- 1 — . + 1 ... + 1 + 4; ... + 1 + 2 + 6 , + 2 + 1 + 2 + 5 + 1214- 5 + 1 + 4 + 6 + 12 + 241+ 12 + 1 « • • • • • • • • + 4 + 17; ... + 1 + 3 • • • . 3 + 7!+ 18 + 46 + 12 + 1 + 3 + 5 + 6 + 2 + 11 + 1«!+ 39 + 84+31 + 1 + 1 + 2- • • « + 16 + 30 • • • + 10 + 32 + 20 1 + 55 + 140 -r 30 1 ^ 4 + 14 + 53 + 150 + 114 + 246 + 80 1 + 1 + 1 + 8 + 6 + 28 + 13 + 15 + 56 + 51 + 108 + 20 + 70 !+ 95 + 315 + 660 + 210 + 1680 ' + 560 4 + 56 + lt)8 + 336 + 280 + 420 1 + 840 ■| 1'3» i 12«3 he'd P23 iy'cd 1«3 bH 2* c* P2» 6V P2« 6V 1«2 6«c (8) 1 + 1 (71) 1 ' - + I + 8 (62) 1 1 ; + + 1 + 6 + 28 (53) + 1 4 + 15 + 56 (4') i+ 1 + 2 + 6 + 20 + 70 (61») + 1 1 • • • + 2 + 13 + 56 (521) + 1 + 5 • • • + 3 + 14 + 51 + 168 (♦31) + 1 + 3 + 10 + 4 + 11 + 32 + 95 + 280 (42') + 1 + 2 + 7 + 20 + 30 + 6 + 18 + 53 + 150 + 420 (3»2) + 2 + 5 + 12 + 12+31 + 80 + 210 + 560 (31') • • • + 2 • • • + 3 + 16 • • • + 6 + 30 + 108 + 336 (421') + 5 + 18 + 55 + 12 + 39 + + 114 172 + 315 + 840 (3'1') + 4 + 12 + 30 + 80 + 28 + ()8 + 440 + 1120 (32'1) + 12 + 24 + 58 + 140 + 48 + 117 + 284 + 690 + 1680 (2«) + 28 + 48 + 108 + 240 + 90 + 204 + 468 + 1080 + 2620 (iV) + 6 + 12 + 46 + 140 + 24 + 84 + 246 + 660 + 1680 (321') 4- 30 + 58 + 141 + 258 + 340 + 108 + 258 + + 612 1008 + 1440 + 3360 (2'!') + 68 + 117 + 570 + 204 + 453 + 2250 + + 5040 6720 (31') + 80 + 172 + 140 + 340 + 800 + 240 + 570 + 1320 + 3000 (2'1*) + 284 + 612 + 1320 + 468 + 1008 + 2172 + 4680 + 10080 (21«) + 440 + 690 + 1440 + 3000 + 1080 + 2250 + 4680 + 9720 + 20160 (1») + 1120 + 1680 + 3360 + 6720 ' + 2520 + 5040 + 10080 + 20160 ^ 40320 147] OF THE ROOTS OF AN EQUATION. 427 VIII (6). Runs on iattk. (8) (71) (62) (53) (4») (6P) (521) (431) (42') (3'2) (51') (421») (3n«) (32*1) (2') (41^) (321*) (2»1») (31») (2'1*) (21') (1") 8 • 17 bh 26 P6 - b + 1 + 2 + 8 + 4 - 1 -3 - 9 - 2 - 5 + 1 + 4 + 5 + 5 • • • - I -5 • • • + 1 35 df + 8 -8 -8 + 7 4 + 8 + 1 + 1 + 8 -7 -3 9 + 3 + 6 - 2 + 3 -3 + 1 125 w P5 ^/ + 8 - 1 - 2 -3 - 4 + 1 4* e« + 4 -4 134 hde 2M 1«24 IH 23« - 8 + 8 + 8 - 16 + 9 16 - 8 + 21 -8-8 + 1' + 8 + 2! + 2 + .3| 7 ... ' + 4 - 1 - 5 -3+5 ... 1 — 1 — 2 ... + 1 1 1 + 8 - 1 - 8 + 9 + 8 -4 - 10 + 8 - 8 + 4 + 4 -4 -4 + 6 + 4 + 8 -8 - 4 + 16 - 6 + 8 - 8 - 8 + 1 + 1 + 8 - 4 -2 - 4 • • • - 9 + 4 - 4 - 4 + 8 - 8 + 4 8 + 1 + 2 - 3 - 9 + 4 -- ]6 -16 + 9 + 4 - 8 + 3 - 10 + 11 + 9 + 16 10 + 4 + 2 • • • + 10 - 1 8 - 8 + 8 + 8 - I - 4 4 • • • + 4 - 2 + 2 -2 + 1 - 2 • • • + 2 + 5 + 4 - \ 1 + 8 _ 2 + 3 - 1 - 4 ■ • • • • • • • • + 1 + 4 -- 4 + 24 - 10 - 6 H- 11 + 4 - 1 + 1 + 12 + 24 - 5 - 9 - 1 + 2 - 2 - 17 • • • - 3 - 4 + 1 + 1 1 1 1 1 1 1 1 •f 2 - 8 - '2 + 1 + 2 + 2 • • • - 3 -32 + 11 + 8 + 1 . i-16 + 9 - 4 + 8 - 1 - 2 + 20 - 6 + 1 1 - 8 1 + 1 ^2 1'3« 6W 12«3 hi?d 1»23 1»3 h'd + 8 - 1 -2 • • • • • • + 1 2* c* + 2 - 2 + 2 -2 + 1 P2» P2» 1«2 -8 + 1 («) + 12 + 24 32 16 + 20 + 1 (71) - 5 -17 + 11 + 9 - 6 (62) - 9 • • • + 8 4 + 1 (53) + 3 + 6 - 3 + 1 (61) (521) + 2 + 5 4 • • • - 5 1 + 5 1 - 1 - 3 + 1 (431),- 2 + 1 _ — (42-') ' + 1 i • 1 1 ! 1 1 ! I 54—2 428 A KEHOIR ON THE SYMMETRIC FUNCTIONS [147 IX (a). Runs on to p. 430. II 9 • - 1 18 6i 27 ch 1'7 - 1 - 2 - 15 -72 35 dg - 1 • • • - 5 - 21 -84 125 hcg 1»6 6V 45 135 hdf 2«5 1»25 1*5 14» <9) (81) (72) (63) (64) (71') (621) (531) (4n) 1 I (52') (432) (3') * - 1 -9 - 1 - 7 -36 (61') (521') (431«) (42'1) (3'21) (32') - 1 (51*) 1 • • • (421») 1 4 • • ■ (3n«) - 1 2 6 • • • (32»1») - 1 - 2 5 - 12 - 2 (2*1) - 1 - 4 - 6 - 12 - 24 - 9 (41-) - 1 • • • • • • • • • 5 - 21 • • ■ (321«) - 1 - 3 • • • - 4 - 9 - 23 - 58 - 6 (2'1') - 3 - 6 - 3 - 15 - 24 - 51 - 108 - 24 1 (31«) - 6 - 19 • • • - 15 - 30 - 81 - 204 - 20 (2»1») - 17 - 36 - 10 - 50 - 81 - 172 - 366 - 70 (21') - 70 -147 - 35 -161 -252 - 525 - 1092 -210 (!•) -252 -504 - 126 -504 -756 -1512 -3024 -630 147] OF THE ROOTS OF AN EQUATION. 429 II (81 (72; (63) (54; (71') (62i; (53i; (4n; (52«) (432) (3») (61») (521«) (431«) (42«i; (3«2i; (32' (51*) (42P) (3*1»J (32'1'^ (2*1 (4r (321*) (2»1») (31') (2»1») (210 (!•) 234 cde 1»34 b^de 12H be'e 1»24 b^ce P4 b^e 3» 123» 6ccP P3» b^d" 2»3 c'd 1«2«3 • 1 1 2 1 • • • 2 1 3 3 8 1 3 6 6 15 1 • • • • • • • • • • • « • • • 1 5 • • • • • • 2 • • • 5 1 3 10 • • • 2 6 7 19 1 2 7 20 • • • 5 17 - 12 36 1 2 5 - 12 30 3. - 13 30 - 27 65 3 7 - 12 - 27 60 6 - 27 64 - 51 - 120 ■ • • • • ■ • • • 4 21 • • • • • • 6 • • • 12 • • • 3 7 - 25 75 • • • - 12 42 - 27 85 3 6 - 17 - 42 - 110 6 - 30 72 - 64 - 152 8 - 19 - 36 85 - 200 - 15 - 65 - 152 - 120 - 281 - 22 - 48 - 78 - 168 - 360 - 36 136 - 300 - 234 - 516 • • • - 10 - 20 - 75 - 225 • • ■ - 30 - 110 - 60 - 200 - 22 54 - 101 - 241 - 570 - 36 - 158 - 372 - 282 - 656 - 60 - 129 - 213 - 459 - 990 - 93 - 333 - 720 - 555 - 1203 - 60 - 155 - 270 - 645 - 1500 - 90 - 390 - 920 - 1740 - 660 - 1530 - 166 - 350 - 565 - 1200 - 2550 - 240 - 820 -1320 - 2800 - 455 - 945 - 1470 -3780 -3045 - 6300 - 630 -2030 - 4200 -3150 - 6510 -1260 -2520 -7560 15120 - 1680 -5040 - 10080 -7560 - 15120 4S0 A MfcMOIR ON THE SYMMKTRIC FUNCTIONS [147 II 1*23 h'cd P3 1 12* 6c* 1*2» 6V 1»2« 6V r2 6» (9) 1 (81) 1 9 (72) 1 7 36 (63) 1 5 21 84 (54) 1 3 10 35 126 (7P) 1 • • • • • • 2 15 - 72 (621) 1 6 • • • 3 17 70 252 (531) 4 6 15 4 15 50 161 504 (4n) 20 9 6 2i 70 210 - 630 (52«) 9 30 24 '- 81 252 756 (432) 22 60 90 22 60 !- 165 455 - 1260 (3«) - 36 36 93 - 240 630 - 1680 (6P) 3 19 • • • 6 36 147 504 (52 P) 23 81 12 51 - 172 525 - 1512 (43P) 54 - 155 48 - 129 - 350 945 - 2620 (42«1) - 101 - 270 78 - 213 i- 565 - 1470 - 3780 (3«21) - 158 390 - 136 - 333 i- 820 2030 - 5040 (323) - 282 - 660 - 234 - 555 - 1320 - 3150 - 7560 (51*) 58 - 204 - 645 24 - 108 1- 366 - 1092 - 3024 (42P) - 241 - 168 459 - 1200 - 3045 - 7560 (3»1») - 372 - 920 - 300 - 720 - 1740 - 4200 - 10080 (32n«) - e.'ie 1530 516 - 1203 '- 2800 - 6510 - 15120 (2*1) 1140 - 2520 - 906 - 2016 1- 4500 - 10080 - 22680 (4P) - 570 - 1500 - 360 990 - 2550 - 6300 - 15120 (321*) 1516 - 2610 - 3480 - 1140 - 2610 - 5940 - 13440 - 30240 (2»P) - 5670 - 2016 - 4383 - 9540 - 20790 - 45360 (3P) - 3480 - 7800 - 2520 - 5670 1- 12600 - 27720 - 60480 (2»P) - 5940 - 12600 - 4500 - 9540 - 20220 - 42840 - 42840 - 90720 (2r) - 13440 - 27720 - 10080 - 20790 - 88200 - 181440 (!•) - 30240 - 60480 - 22680 - 45360 - 90720 - 181440 - 362880 1 47] OP THE ROOTS OF AN EQUATION. 431 IX ijb). Runs on to p. 432. — 9 • J 18 bi 27 ch V7 35 d(/ 125 beg 1'6 b'g + 9 - 1 -2 -3 -9 + 1 + 3 + 4 + 5 + 2 + 5 • • • - 1 -4 -5 -5 • • • • • • + 1 + 5 • • • • « • • • • - 1 45 135 bdf 2«5 -9 1«25 b'cf 1*5 hV -9 + 1 + 2 + 3 + 4 - 1 -3 -4 • * • -2 • • • • • • + 1 + 4 • • • • • • • • • • • • - 1 14« b^ 234 cde 1*34 (9) - 9 + 9 + 9 - 9 + 9 - 18 + 9 - 9 - 18 + 27 - 9 - 18 + 27' (81) + 9 - 1 - 9 + 1 - 9 + 10 + 10 + 9 - 11 + 5 + 18 + 4 - 11 (72) + 91- 9 + 5 + 2-9 + 4 - 9 + 18 -5 + 9 - 6 + 9 -20 (63) + 9 - 9 - 9 + 9 + 9 • • • -. 9 • • • - 9 + 9 • • • - 9' (54) + 9 - 9 - 9 - 9 + 9 - 9 + 18 + 11 - 2 - 1 - 7 -11 - 2 + 13 + 11 (71') + 1 + 2 - 1 + 9 - 3 + 9 - 10 -2 + 4 - 5 - 11 (621) -18 1+10 + 4 - 3 • • • - 8 + 18 - 10 - 4 -8 + 11 - 14 - 4 + 2 + 13i (531) - 18'+ 10 + 18 - 10 • • • + 9 - 10 - 2 - 5 + 15 + 6 - Tj (4«1) - 9 + 5 + 9 - 5 - 14 - 11 + 6 + 1 • • • + 6 -3 + 2 + 6 • • • -6 + 3 • • • -2 + 2 - 1 + 5 + 2 - 5 (52') -9+9 - 5 - 2 + 9 - 4 - 1 - 8 + 2 + 6 - 6 + 1 + 6 • « • + 1 (432) -181+18 + 4 - 11 • • • - 4 - 2 + 2 - 8 + 3 + 5 (3'; - 3 + 3 - 1 + 3 - 2 - 3 - 6 + 3 + 3 + 3 • • • - 3 • • • (61') + 9 + 1 - 3 - 9 + 3 - 9 + 4 - 4 + 5 - 5 (i521») + 27 - 11 - 6 + 4 + 11 + 5 + 11 - 7 + 13 + 3 + 15 - 15 - 1 - 5 + 1 (m') + 27 - 11 - 20 + 1 - 13 - 9 - y + 13 + 12 - 5 - 2 - 4 + 2 + 1 + 3 - 5 + 1 + 2 (+-«1) + 27 - 19 + 1 + 2 - 11 1 (:V''21) + 27' - 19 - 9 + 12 + 2 + 18 . — 7 1 - 7 • • • + 3 - 1 (32») + 9 + 5 - 3 + 3 - 2 + 1 • • a - 1 (51') - 9 + 1 + 2 - 1 - 5 - 3 + 4 - 4 + 4 1 (421») -36 - 18 + 12 + 8 + 12 - 14 - 4 + 1 - 1 (3n») + 6 + 11 - 6 - 3 + 1 + 4 - 2 + 2 - 1 (32n») -54 + 30 + 5 — 5 - 9 - 9 + 3 + 4 - 1 - 1 1 (2'1) - 9 + 7 • • • • • • 1 . (41») + 9 + 45 - 1 - 2 + 1 - 3 + 3 1 (321*) - 13 - 10 + 6 + 3 - 1 (2'1») + 30 - 14 + 5 • • ■ - 1 1 (31*) - 9!+ 1 + 2 - 1 - 1 " (2'1») - 27 + 7 1 (210 + 9 - 1 ! 1 1 (!•) - 1 1 432 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 __ 12«4 bi^e P24 b^ce P4 b'^e + 9 - 1 -2 -3 • • • + 1 + 3 - 1 3» (P -3 + 3 + 3 -6 + 3 -3 + 3 + 3 -3 -3 + 3 - 1 123» bccP P3» b^(P 2»3 + 9 -9 + 5 -3 + 1 + 2 - 2 + 2 -1 1«2>3 1*23 b*cd 1*3 b*d -9 + I + 2 • • • « • • - 1 12* be* -9 + 7 -5 + 3 - 1 1«2» 6V 1»2» V2 b'c + 9 -1 I* -1 (9) + 27 -36 + 27 - 18 -54 + 45 + 30 -27 (81) - 19 + 12 -19 + 6 + 30 -13 -14 + 7 (72) + 1 + 8 - 13 + 11 + 5 -10 + 5 - 1 (63) - 9 + 12 + 18 - 3 - 9 + 3 - 1 (54) + 3 - 4 - 7 - 2 + 4 • • • (7P) + 5 - 5 + 12 - 6 - 9 + 6 (621) + 12 - 14 + 1 - 7 + 1 + 4 - 1 (531) - 2 - 4 + 2 - 1 (in) + 1 • • • + 3 • • • (52«) - 6 + 2 + 3 - 1 (432) + 2 • • a - 1 (3») • • • • • • (61') - 5 + 5 (52P) + 3 - 1 (43 1«) - 1 i 1 • ! 1 147] OF THE ROOTS OF AN EQUATION. 433 X (a). Runs on to p. 436. II 10 k 19 28 ci P8 hH 37 dh 127 hch 1»7 6»A 46 ^9 136 hdg 2»6 P26 h'cg P6 h'g 5» (10) (91) (82) (73) (64) (5') (8P) (721) + 1 (631) (541) (62') (532) (4'2) (43») (71') (62 1») (53P) • (4n') (52=1) (4321) (3'1) (42') (3»2') (6P) (52P) (431') , (42n») (3»2P) (32'1) (2») + 1 (5P) + 1 • • • (421«) + 1 + 2 + 17 + 90 + 1 + 4 • • • (3'P) -f 1 + 2 + 6 • • • (32'1») + 1 + 2 + 5 + 12 • • • (2*1') + 1 + 4 + 6 + 12 + 24 + 2 (41») + 1 • • • • • • ■ • • -f 6 + 25 • • • (321») + 1 + 3 • • • -f 5 + 11 + 28 + 70 • • • (2«1') + 1 + 3 + 6 + 4 + 19 + 30 + 63 + 132 + 6 (310 • • • + 7 + 22 • • • + 21 -f 42 -f 112 + 280 • • • (2'1') + 1 -f 8 + 45 + 6 + 20 + 42 + 15 + 72 + 115 + 242 + 510 + 20 (21») + 1 + 10 + 28 + 92 + 192 -f 56 + 252 + 392 + 812 + 1680 + 70 (1") + 120 -f 360 -f 720 + 210 + 840 + 1260 + 2520 + 5040 + 252 : C. II. 55 434 A MEMOIB OK THE STMMETBIC FUNCTIONS [147 II 145 235 cdf 1«35 12»5 hi?f 1»25 hhf P5 24« 1»4» 6V 3>4 d^e 1234 hcde (10) (91) (82) (73) (64) (5') (81') (721) (631) (541) (62') (632) (4«2) (43«) (71') (621') (531') (4«1') (52*1) (4321) + 1 (3'1) + 1 + 3 (42») + 1 « • • + 3 (3»2») + 1 + 2 + 2 + 8 (61') + 1 • • • • • • • • • • • • (521») + 1 + 5 • • • • • • • • « m • • (431') + 1 + 3 -f 10 • • • • • • • • • + 3 (42'1») + 1 + 2 + 7 + 20 • • • + 2 • • • + 8 (3'21') -f 1 + 2 + 5 + 12 + 30 + 2 + 4 + 5 + 21 (32*1) + 1 + 3 + 7 + 12 + 27 + 60 + 7 + 16 + 12 + 49 (2») + 5 + 10 + 20 + 30 + 60 + 120 + 20 + 45 + 30 + 110 (51") • • • • • • • * • ... + 5 + 26 • • • • • • • • • • • • (421') • • • • • • -f 4 + 9 + 32 + 95 • • • + 6 • • • + 22 (3»1*) • • • + 4 -f 8 + 22 + 54 -f 140 + 6 + 12 + 12 + 56 (32»1') -f 3 + 11 -f 26 -f 48 + 112 -f 260 + 480 + 18 + 42 + 31 + 128 (2n') + 14 -f 32 + 68 + 108 + 228 + 53 + 114 + 80 + 284 (41«) • • • • • • + 15 + 30 + 111 + 330 • • • + 20 • • • + 60 (321») + 10 + 35 4- 85 + 156 + 368 + 860 + 50 + 120 + 80 + 335 (2'1<) + 42 + 99 + 210 + 339 + 720 + 1530 + 144 + 306 + 213 + 735 (310 + 35 + 105 -f 266 + 462 + 1092 + 2620 + 140 + 360 + 210 + 876 (2»1«) + 130 + 296 + 622 + 990 + 2082 + 4380 + 400 + 840 + 570 + 1900 (21') + 406 + 868 -f 1792 + 2772 + 5712 + 11760 + 1120 + 2310 + 1540 + 4900 (1") + 1260 + 2520 + 5040 + 7560 + 15120 + 30240 + 3150 + 6300 + 4200 + 12600 147] OF THE ROOTS OF AN EQUATION, 435 P34 2»4 1«254 6Vc 1*24 1«4 13' 223» 1>23« 1*3» h'd^ 12»3 h&d • -f 1 + 1 • • • + 1 + 4 + 3 + 1 + 2 + 6 + 7 + 1 -f 2 + 5 + 12 + 12 + 1 • • • • • • • • • • • • + 1 + 6 • • • • • • + 2 • • • + 1 + 4 + 15 • • • + 2 + 8 + 7 + 1 -f 2 -f 6 + 20 + 2 + 4 + 12 + 16 + 1 ■ • • -f 2 + 9 + 30 • • • + 5 + 22 -f 12 + 3 + 3 + 8 + 22 + 60 + 3 + 8 + 21 + 56 + 49 -f 87 + 6 + 6 + 15 + 36 + 90 -f 10 + 18 + 42 + 96 + 10 + 6 + 18 + 48 + 120 -f 6 + 15 + 42 + 115 + 87 + 18 + 15 + 34 + 78 + 180 + 18 + 34 + 80 + 188 + 156 • • • • • • • • • + 4 -f 25 • • • • • • • • • + 6 • • • + 3 • • • + 7 + 32 + 76 + 111 • • • • • • + 12 + 54 + 27 + 9 + 10 + 27 + 215 + 6 -f 18 + 48 + 132 + 112 + 27 + 18 + 54 + 149 -f 390 + 15 + 34 -f 99 + 270 -f 198 + 48 + 42 + 99 + 236 + 570 -f 42 + 80 + 186 + 436 + 358 + 112 + 87 + 198 + 450 + 1020 + 87 -f 156 + 358 + 820 + 645 + 240 + 180 + 390 + 840 + 1800 + 180 + 310 + 680 + 1500 + 1170 + 10 • • • + 20 + 95 -f 330 • • • • • • -f 30 + 140 + 60 + 76 + 48 -f 149 + 416 + 1095 + 36 + 78 + 236 -f 650 + 450 + 132 + 115 + 270 + 650 + 1580 + 96 -f 188 + 436 + 1032 + 820 + 294 + 228 + 523 + 1196 + 2730 + 210 + 370 + 844 + 1920 + 1479 + 612 + 468 + 1008 + 2172 + 4680 + 444 + 740 + 1604 + 3480 + 2688 + 215 + 120 + 390 + 1095 + 2850 + 90 + 180 + 570 + 1580 + 1020 + 775 + 585 + 1340 + 3050 + 6900 + 510 + 880 + 2000 + 4520 + 3390 + 1566 + 1194 + 2547 + 5436 + 11610 + 1092 + 1776 -h 3792 + 8100 + 6180 + 2030 + 1470 + 3360 + 7560 -f 16800 + 1260 + 2100 + 4760 + 10640 + 7770 + 3990 + 3015 + 6330 -f 13290 + 27900 -f 2700 + 4280 + 8980 -f 18840 + 14220 + 10080 + 7560 + 15540 + 31920 + 65520 + 6720 + 10360 + 21280 + 43680 + 32760 + 25200 + 18900 + 37800 + 75600 + 151200 4- 16800 -f 25200 -f 50400 -f 100800 + 75600 55-2 436 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 II i l»2«3 9c'd P23 b^cd V3 2» 2»3> b^c* P2» 6V 1«2» 6V 1»2 110 6" (10) 4- 1 (91) 4- 1 4- 10 (82) 4- 1 4- 8 4- 45 (73) • 4- 1 4- 6 4- 28 4- 120 (64) 4- 1 4- 4 4- 15 4- 56 4- 210 (5') 4- 1 4- 2 4- 6 4- 20 + 70 4- 252 (81') 4- 4- 1 t • • • • • • • • • 4- 2 4- 17 4- 90 (721) + 1 • • • • • • 4- 3 4- 20 4- 92 4- 360 (631) + 1 + 5 4- 21 • • • 4- 4 4- 19 4- 72 + 252 4- 840 (541) + 3 + 10 4- 35 4- 5 4- 14 4- 42 4- 130 4- 406 4- 1260 (62») + 2 ■+ 11 4- 42 • • • 4- 6 4- 30 4- 115 4- 392 4- 1260 (532) + 11 + 35 4- 105 4- 10 4- 32 4- 99 4- 296 4- 868 4- 2520 (4*2) + 18 + 50 4- 140 4- 20 4- 53 4- 144 4- 400 4- 1120 4- 3150 (43') + 31 + 80 4- 210 4- 30 4- 80 4- 213 4- 570 4- 1540 + 4200 (7P) • • • + 3 4- 4- 22 112 • • • • • • 4- 6 4- 42 4- 192 4- 720 (62 1») + 5 + 28 • • • 4- 12 4- 63 4- 242 4- 812 4- 2520 (sai*) + 26 + 85 4- 266 4- 20 4- 68 4- 210 4- 622 + 1792 4- 5040 (4'1») + 42 + 120 4- 350 4- 45 4- 114 4- 306 4- 840 4- 2310 4- 6300 (52n) + 48 + 156 4- 462 4- 30 4- 108 4- 339 4- 990 4-' 2772 4- 7560 (4321) + 128 + 335 4- 875 4- 110 4- 284 4- 735 4- 1900 4- 4900 4- 12600 (3'1) + 210 + 510 4- 1260 4- 180 + 444 4- 1092 4- 2700 4- 6720 4- 16800 (42') + 228 + 585 4- 1470 4- 180 4- 468 4- 1194 4- 3015 4- 7560 4- 18900 (3»2'') + 370 + 880 4- 2100 4- 310 4- 740 4- 1776 4- 4280 4- 10360 4- 25200 (61') + 12 + 70 4- 280 • • • 4- 24 4- 132 4- 510 4- 1680 4- 5040 (521») + 112 4- 368 4- 1092 4- 60 4- 228 4- 720 4- 2082 4- 5712 4- 15120 (43P) + 294 + 775 4- 2030 4- 240 4- 612 4- 1566 4- 3990 4- 10080 4- 25200 (42n») + 523 + 1340 4- 3360 4- 390 4- 4- 1008 4- 2547 4- 6330 4- 15540 4- 37800 (3»21') + 844 + 2000 4- 4760 4- 680 1604 4- 3792 4- 8980 4- 21280 4- 50400 (324) + 1479 + 3390 4- 7770 4- 1170 4- 2688 4- 6180 4- 14220 4- 32760 4- 75600 (2') + 2580 + 5700 4- 12600 4- 2040 4- 4530 4- 10080 4- 22500 4- 50400 4- 113400 (51') + 260 4- 860 4- 2520 4- 120 4- 480 4- 1530 4- 4380 4- 11760 4- 30240 (121*) + 1196 + 3050 4- 7560 4- 840 1500 + 2172 4- 5436 4- 13290 4- 31920 4- 75600 (3»1*) 4- 1920 4- 4520 4- 10640 4- 4- 3480 4- 8100 4- 18840 4- 43680 4- 100800 (32n>) + 3358 4- 4- 7610 12720 4- 4- 17220 4- 2580 ; 4- 5844 4- 13212 4- 29820 4- 67200 4- 151200 (2n») + 5844 27720 4- 4530 4- 9876 4- 21564 4- 47160 4- 103320 4- 226800 (41«) + 2730 4- 6900 16800 4- 1800 4- 4680 + 11610 4- 27900 4- 65520 4- 151200 (321*) + 7610 4- 4- 17000 28260 4- 37800 1 4- 5700 + 12720 4- 28260 4- 62520 4- 137760 4- 302400 (2»1*) -f 13212 4- 60480 4- 10080 4- 21564 + 46152 4- 98820 4- 211680 4- 453600 (31') + 17220 4- 37800 4- 82320 4- 12600 4- 27720 4- 60480 4- 131040 4- 282240 4- 604800 (2n») + 29820 4- 62520 4- 4- 4- 131040 + 22500 4- 47160 4- 98820 4- 207000 + 433440 4- 907200 (21') 67200 4- 137760 302400 282240 4- 50400 4- 103320 4- 211680 4- 433440 4- 887040 4- 1814400 (V) + 151200 4- 604800 4- 113400 4- 226800 4- 453600 4- 907200 4- 1814400 1 + 362880C ii 147] OF THE BOOTS OF AN EQUATION. 437 10 k 19 28 ci P8 bH £(6). 37 dh Run 127 bch -20 s on 1 P7 ko p. - 46 139. 136 bdg 2>6 P26 b^'cg 4- 30 - 12 1*6 b'g 5» 145 be/ (10) 10 + 10 + 10 - 10 + 10 4- 10 4- 10 -20 - 10 -10 + 5 20 (91) + 10 - 1 10 4- 6 4. 1 - 10 4- 11 - 1 - 2 - 10 4- 11 4- 10 4- 1 4. 2 — 5 + 11 (82) + 10 - 10 4- 2 - 10 4- 4 10 4- 20 - 6 - 6 — 5 + 20 (73) + 10 - 10 - 10 4- 10 4- 11 - 1 - 3 - 10 - 1 4- 10 - 9 4- 3 — 5 + 20 - 4 (64) + 10 - 10 - 5 - 10 4- 10 - 10 4- 20 - 10 4- 14 - 4 - 2 - 6 4- 4 4- 5 10 (5*) + 5 - 5 4- 5 - 5 4- 10 5 5 4- 10 4- 5 - 15 + 5 - 15 (8P) 10 + 1 ^ ^ - 1 4- 10 - 3 4- 1 4- 10 - 11 - 2 - 4 + 4 - 1 4- 5 - 11 -31 7 (721) - 20 + 11 4- 4 - 3 - 1 - 8 4- 3 + 20 - 10 4- 11 - 3 - 4 4- 10 (631) - 20 + 11 + 11 4- 20 + 20 - 11 - 1 -10 + 4 4 - 4 - 8 4- 15 4- 10 15 (541) - 20 - 11 - 2 4- 20 -31 + 11 - 4 - 7 - 8 4- 18 4- 6 - 5 4- 23 (62*) - 10 + 10 - 6 + 10 - 4 4- 2 - 2 - 8 • • ■ - 2 - 5 + 5 - 8 (532) - 20 4- 20 4- 4 4- 2 - 12 1 - 3 4- 5 4- 20 - 19 - 4 4. 10 4. 15 — 15 + 10 (4«2) - 10 + 10 - 6 4- 10 - 12 + 6 4- 3 14 4- 4 - 6 • • • + 5 4- 4 (43«) 10 + 10 - 1 + 10 - 10 - 11 4- 1 - 2 4. 13 - 4 - 3 • • • 4- 5 - 8 (7P) + 10 2 -6 4- 1 4- 4 - 3 4. 3 - 1 - 10 4. 4 4- 2 - 4 4. 1 — 5 4. 11 (62P) + 30 - 12 9 + 11 - 4 - 6 4- 15 4. 6 15 4- 4 — 15 4- 18 (53P) + 30 - 12 -22 - 11 4- 2 4. 12 - 9 4. 13 5 - 6 4- 15 4- 10 - 19 4- 5 4- 10 - 12 (4«P) + 15 - 6 4- 6 - 15 4- 17 - 6 4- 9 - 3 - 1 + 1 • • • 4- 5 - 8 (52«1) + 30 -21 4- 5 - 9 4- 12 - 5 - 18 4- 18 + 4 -17 4- 5 4- 10 - 1 (4321) + 60 -42 -28 4- 26 4. 3 4- 21 - 12 4- 12 - 15 - 8 4- 7 • • • — 5 • • • (3'1) + 10 7 - 10 4- 7 + 11 - 4 • • • 4- 2 - 7 • • • 4- 4 • • • • • • - 5 4- 5 (42») + 10 - 10 4- 6 4- 2 - 10 4- 4 - 2 + 10 - 4 4- 2 • • • — 5 • • • (3«22) 4- 15 - 15 + 1 + 13 + 13 4- 1 4- 7 4- 6 - 7 • • • - 9 4- 3 4- 2 • • • • • • 4- 5 - 1 (6P) - 10 4- 2 + 8 - 1 + 3 - 3 + 1 4- 4 - 4 - 2 4- 4 - 1 4- 5 - 5 (521») - 40 - 5 + 12 - 14 - 16 - 23 + 5 4. 16 -19 - 8 4. 19 - 5 — 5 4- 1 (43P) - 40 + 24 - 13 - 9 -21 4- 12 + 6 - 8 4- 5 • • • - 1 — 5 4- 5 (42212) - 60 4- 33 + 33 4- 4 4- 18 + 9 - 12 4- 3 + 8 4 4- 5 - 1 (322P) - 60 4- 28 -24 4. 9 • • • • • • 4- 6 - 4 • • • 4- 5 - 3 (32n) - 40 + 31 - 8 - 2 - 7 - 2 4- 5 • • • 4. 8 - 3 • • • ■ • • — 5 4- 1 (2») - 2 + 2 • • • 4. 2 • • • • • • - 2 • • • • • • • • • 4- 1 (5P) + 10 - 1 2 + 1 - 3 + 3 - 1 - 4 4. 4 4- 2 - 4 4- 1 (42P) + 50 - 14 - 10 + 6 -15 4- 17 - 6 4- 4 1 2 4- 1 (3«P) + 25 - 7 - 13 4. 7 4- 3 4- 12 - 1 • • • 4- 2 - 2 4- 1 (32n») + 100 46 - 12 4. 14 - 5 • • • - 4 4. 1 (2n') + 25 - 16 + 9 4- 2 • • ■ - 4 + 3 • • • • • • 4- 1 (4P) - 10 + 1 - 1 - 3 + 1 (32P) - 60 4- 15 + 12 - 7 - 3 4- 1 (2'P) - 50 + 20 6 • • • 4. 1 (3F) + 10 1 - 2 + 1 (2n«) + 35 - 8 4- 1 (2P) - 10 4. 1 1 1 (PO) + 1 438 A MEMOIR ON THE SYMMETRIC FUXCTIONS [147 — 235 cdf 1>35 hHf + 30 12 12«5 6cy P25 1»5 24» C6* 1H» 6V 3«4 1234 bcde + 60 -42 P34 b^de 2»4 c'e 1»2H 1*24 b^ce b^e (10) -20 + 20 + 30 -40 + 10 - 10 + 15 - 10 + 10 -40 + 10 -60 + 50 -10 (91) -21 + 13 - 1 + 10 + 2 - 6 + 13 -10 + 33 -14 + 1 (82) + 4 -22 + 2 + 8 - 2 - 11 + 10 -11 - 2 + 5 -28 + 24 + 12 + 6 + 4 - 10 + 2 (73) - 1 - 9 - 9 + 12 - 3 + 10 -15 + 3 + 12 - 10 + 18 - 15 + 3 (64) + 20 - 6 - 18 + 16 4 - 14 + 5 + 9 + 5 - 8 + 10 - 5 - 12 + 4 • • • • • • - 1 - 3 (5^) - 15 + 10 + 10 - 5 • • • - 5 - 5 + 5 • • • (8P) - 12 + 12 + 5 - 5 + 1 + 3 - 6 + 6 - 10 + 26 -13^ + 2 - 9 + 6 (721) - 3 + 13 + 12 - 14 - 12 + 17 + 1 + 21 - 16 + 4 -23 + 17 (631) - 19 + 15 + 18 - 19 + 4 + 4 + 4 - 3 + 13 - 15 + 5 • • • • • • + 3 - 1 (541) + 10 - 12 - 1 + 1 • • • - 8 8 • • • + 5 - 1 • ■ • (62») - 4 + 10 - 4 + 2 + 4 - 8 + 2 + 10 - 1 - 4 + 1 ; 2 - 8 + 5 + 4 - 3 • • • - 4 + 8 - 2 (532) + 17 - 13 + 5 • • • - 12 + 1 3 + 3 - 6 - 1 • • • + 4 - 2 ■ • • (4«2) - 12 + 2 • • • • • • + 5 + 19 • • • • • • + 2 + 2 + 6 - 6 + 2 - 2 • • • • • • (43«) + 1 - 3 + 3 - 1 • • • • • • • • • • ■ • + 1 (7P) + 5 - 5 - 5 - 17 - 1 - 4 + 3 12 + 7 + 6 - 2 + 9 - 6 (621«) + 15 - 4 - 19 + 1 - 3 - 3 - 1 + 2 - 2 - 4 + 1 (53 1«) + 3 + 2 1 + 2 + 4 - 2 + 1 (4n«) -f 1 + 2 - 1 • • • - 3 + 3 + 3 - 3 • • • + 1 (52n) - 13 + 2 + 9 - 3 - 3 • • • • • • • • • • • • - 5 + 1 + 2 - 1 - 3 + 3 - 3 + 1 (4321) + 5 + 4 + 4 - 3 + 1 + 1 (3'1) - 1 • • • • • • - 2 - 2 + 1 • • • (42') + 4 - 2 ■ • • + 1 (3>2*) - 2 - 5 + 5 - 1 - 2 + 1 • • • • • • (61*) + 5 + 5 + 1 (52 P) 1 - 3 (43P) 2 + 1 (42n') + 1 (3»21>) 147] OF THE ROOTS OF AN EQUATION. 439 13» hd^ 2'3« c^d^ 1«23« h^cd^ 1*3« h*d^ 12»3 P2«3 P23 h^cd r3 Vd 2» c» -2 + 2 -2 + 2 -2 + 1 2>3» +'25 1*2» 6V 1«2» 6V P2 110 + 1 + 10 + 15 -60 + 33 + 25 -40 + 100 -60 + 10 -50 + 35 - 10 - 7 - 15 - 7 + 31 - 46 + 15 - 1 - 16 + 20 - 8 + 1 - 10 + 1 + 28 - 13 - 8 - 12 + 12 - 2 + 9 - 6 + 1 -h 11 + 6 - 24 + 3 - 2 + 12 - 3 • • • - 4 + 1 + 2 - 9 • • • + 2 + 8 - 4 • • • • • • + 1 - 5 + 5 + 5 • • • - 5 • ■ • • • • • • • + 7 + 7 - 21 + 7 - 7 + 14 - 7 + 1 - 4 - 7 + 9 - 1 + 5 - 5 + 1 - 7 + 3 + 6 - 2 - 3 + 1 + 5 - 1 - 3 • • • + 1 + 4 + 2 - 4 + 1 - 1 - 2 + 1 - 2 + 1 + 1 440 [148 148. MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [From the Philosophical Transactions of the Royal Society of London, vol cxLvn. for the year 1857, pp. 703—715. Received December 18, 1856,— Read January 8, 1857.] The Resultant of two equations such as (a, 6, ...50?, y)"* = 0, (p, q, ...5a?, y)'*=0, is, it is well known, a function homogeneous in regard to the coefficients of each equation separately, viz. of the degree n in regard to the coefficients (a, 6, ...) of the first equation, and of the degree m in regard to the coefficients (jd, g, ••) of the second equation; and it is natural to develope the resultant in the form AilP + A'-^.'P' + &c., where A, A\ &c. are the combinations (powers and products) of the degree n in the coefficients (a, 6, ...), P, P', &c. are the combinations of the degree m in the coefficients (p, g, ...), and k, Id, &c. are mere numerical coefficients. The object of the present memoir is to show how this may be conveniently eflfected, either by the method of symmetric functions, or from the known expression of the Resultant in the form of a determinant, and to exhibit the developed expressions for the resultant of two equations, the degrees of which do not exceed 4. With respect to the first method, the formula in its best form, or nearly so, is given in the Algebra of Meyer Hirsch, [for proper title see p. 417], and the application of it is very easy when the necessary tables are calculated: as to this, see my "Memoir on the Symmetric Functions of the Roots of an Equation "(^). But when the expression for the Resultant of two equations is to be calculated without the assistance of such tables, it is I think by far the most simple process to develope the determinant according to the second of the two methods. » Philosophical Transactiont, 1857, pp. 489—497, [147J. 148] MEMOIR ON THE RESULTANT OF A SYSTEM OP TWO EQUATIONS. 441 Consider first the method of symmetric ft^^ctioQS, and to fix the ideas, let the two equations be (a, b, c, cH^x, yY = 0, (jp, q, r "^x, yf = 0. Then writing so that (a, 6, c, d\l, zY = a(l -^ az)(l ^ fie)(l --yzl A = a + fi + y =(1), - f = <^7 - (I'). the Resultant is ip. 3. rSa, 1)' • ip, q, »*M 1)' • (p. q. rJi% lY, which is equal to r' + qr' (a + /5 + f) + pr' (a* + ^ + i*) + pqr (a?/5 + afi* + 0^ + fiy' + '^ + i'a) + Sk.; or adopting the notation for symmetric functions used in the memoir above referred to, this is { r» { + qr' (1) f+pr^ (2) X+q'r (P) j+pqr (21) 1+9* (1*) '+ph- (2») +pq' (2P) {+p'q (2*1) l+P' (2*) . { the law of which is best seen by dividing by r» and then writing ? = [!]. f = [2]. and similarly, ^ = [1']. f=[21].&c; 56 c. ir. 442 MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [148 the expression would then become 1 + [1] (1) + [2] (2) + [V] (V) + [21] (21) + [V] (P) + [2»] (2«) + [21»] (21«) + [2»1] (2n) + [2»] (2'), where the terms vdthin the [ ] and ( ) are simply all the partitions of the numbers 1, 2, 3, 4, 5, 6, the greatest part being 2, and the greatest number of parts being 3. And in like manner in the general case we have all the partitions of the numbers 1, 2, 3, ...mn, the greatest part being n, and the greatest number of parts being tn. The symmetric functions (1), (2), (1*), &c. are given in the Tables (6) of the Memoir on Symmetric Functions, but it is necessary to remark that in the Tables the first coefficient a is put equal to unity, and consequently that there is a power of the coefficient a to be restored as a factor: this is at once eflfected by the con- dition of homogeneity. And it is not by any means necessary to write down (as for clearness of explanation has been done) the preceding expression for the Resultant; any portion of it may be taken out directly from one of the Tables (6). For instance, the bracketed portion + pqr (211 + ?' (1'). which corresponds to the partitions of the number 3, is to be taken out of the Table 111(6). as follows: a portion of this Table (consisting as it happens of consecutive lines and columns^ but this is not in general the case) is be + 8 -1 -1 = d (ii) if in this we omit thd sigh =, and in the outside line write for homogeneity ad instead of d, and in the outside column, first substituting q, p for 1, 2, then write for homogeneity pqr instead of pq, we have ad he pqr + 3 -1 -1 1 viz. pjr X (+3ad — 16c) + g'(— lad), for the value of the portion in queistion; this is equivalent to vqr q^ ^^'''^^'^ , or as it may be mdre convenielitiy Writteti, be in which form it constitutes a part of the expression given in the sequel for the Resultant of the two functions in question; and similarly the remainder of the expres- sion is at once derived from the Tables (b) I. to VI. + 3 -1 -1 148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 443 As a specimen of a mode of verification, it may be remarked that the Resultant qui invariant ought, when operated upon by the sum of the two operations, 3a36 + 263o + c9rf and 2pdq + qdr, to give a result zero. The results of the two operations are originally obtained in the forms in the first and second columns, and the first column, and the second column, with all the signs reversed, are respectively equal to the third column, and conse- quently the sum of the first and second columns vanishes, as it ought to do. \: ^d. \ o*. \ Next to explain the second method, viz. the calculation of the resultant from the expression in the form of a determinant. Taking the same example as before, the resultant is a, 6, c, d a, 6, c, dy 56—2 444 MEMOIR ON THE BBSXTLTANT OF A SYSTEM OF TWO EQUATIONS. [148 which may be developed in the form + 12 . 345 } - 13 . 246 j + 14 . 235 + 23 . 145 - 15 . 234 - 24 . 135 where 12, 13, &c.* are the terms of ( and 123, &c. are the terms of ( + 25 . 134 + 34 . 126 -36 . 124} 4-45 . 123} a, 6, c, d ) a, b, c, d p, q. r viz. 12 is the determinant formed with the first and second oolumns of the upper matrix, 123 is the determinant formed vdth the first, second and third columns of the lower matrix, and in like manner for the analogous symbols. These determinants must be first calculated, and the remainder of the calculation may then be arranged as follows: — \: c^. \ \ r X < J > ^ < \ «v. «♦ 148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 445 where it is to be observed that the figures in the squares of the third column are obtained from those in the corresponding squares of the first and second columns by the ordinary rule for the multiplication of determinants, — ^taking care to multiply the dexter lines (ie. lines in the direction \) of the first square by the sinister lines (i.e. lines in the direction/) of the second square in order to obtain the sinister lines of the third square. Thus, for instance, the figures in the square are obtained as follows, viz. the first sinister line (-h 3, — 1) by (-1, +l)(-2, +1)= 2 + l = + 3, (-1, +1)(+1, 0) = -l + = -l, and the second sinister line (— 1, 0) by (0, -l)(-2, +1) = 0-.1=-1, (0, -l)(^-l, o) = o + o= 0. I have calculated the determinants required for the calculation, by the preceding process, of the Resultant of two quartic equations, and have indeed used them for the verification of the expression as found by the method of symmetric functions; as the determinants in question are useful for other purposes, I think the vahies are worth preserving. Table of the Determinants of the Matrices, and ( ( a, 6, c, d, e ) a, 6, c, d, e a, 6, c, d, e a, 6, c, d, «, f P» tf» r, 8, t ) P> ?» r. s, t P* q> r, «. t P> 9» r, s, t HEHOIB OH THE RESULTANT OF A SYSTEH OF TWO EQUATIONS. [148 arranged in the form adapted for the calculation of the Resultant of the two qiuutic equations (a, b, c, d, e^x, yf = 0, and (p, q, r, s, t'Sx, y)* = 0, viz. 148] MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS. 447 448 MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [l48 -/^■^ ■r % /\ 148] MEMOIR ON THE RE-SHLTANT OF A SV8TGM OF TWO EQUATIONS. 449 The Tables of the Resultants of two uquatious which I have calculated are as 450 MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [148 Table (2, 2). Resultant of (a, 6, c\x, yY, Table (3, 2). Resultant of (a, 6, c d^x, yy, (/>, q. r "^x, yy. Table (4, 2). Resultant of (a, 6, c, d, e^x, yy, (p, 9, r^x, yy. \ \/ 148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. Table (*. 3). Resultant of (a, 6, c, d. e$x, y)*, ip, q, r, sXx, yf. Table (3, 3). Resultant of (a, b. c. d5», sr. (^5, r, .5., y)-. yA '0 '>\<> MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS. Table (4, 4), RfsiiUiint of (a, 6, c, d, e^x; yY. (p. q, r. s, l\x. yy. [148 148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 453 454 [149 149. ON THE SYMMETRIC FUNCTIONS OF THE ROOTS OF CERTAIN SYSTEMS OF TWO EQUATIONS. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 717 — 726. Received December 18, 1856, — Read January 8, 1857.] Suppose in general that ^ = 0, -^ = 0, &c. denote a system of (n — 1) equations between the n variables {x, y, z, ...), where the functions <f>, y^,' &c. are quantics ^e. rational and integral homogeneous functions) of the variable& Any values {xi, yi, Zi,...) satisf}dng the equations, are said to constitute a set of roots of the system; the roots of the same set are, it is clear, only determinate to a common factor pris, Le. only the ratios inter se and not the absolute magnitudes of the roots of a set are deter- minate. The number of sets, or the degree of the system, is equal to the product of the degrees of the component equations. Imagine a function of the roots which remains unaltered when any two sets (o^i, yi, Zi, ...) and (^, y,, z^, ...) are interchanged (that is, when Xi and x^, yi and y,, &c. are simultaneously interchanged), and which is besides homogeneous of the same degree as regards each entire set of roots, although not of necessity homogeneous as regards the different roots of the same set; thus, for example, if the sets are (xi, yi), (x^, y,), then the functions XyX^, XtJ/^ + x^i, y^y^ are each of them of the form in question; but the first and third of these functions, although homogeneous of the first degree in regard to each entire set, are not homo- geneous as regards the two variables of each set. A function of the above-mentioned form may, for shortness, be termed a symmetric function of the roots; such function (disregarding an arbitrary factor depending on the common factors which enter implicitly into the different sets of roots) will be a rational and integral function of the coefficients of the equations, i.e. any symmetric function of the roots may be considered as a rational and integral function of the coefficients. The general process for the investi- gation of such expression for a symmetric function of the roots is indicated in Pro- fessor Schlafli's Memoir, "Ueber die Resultante eines Systemes mehrerer algebraischer 149] ON THE SYMMETRIC FUNCTIONS OF THE ROOTS, &C. 455 Gleichungen," Vienna Transactions, t. iv. (1852). The process is as follows: — Suppose that we know the resultant of a system of equations, one or more of them being linear; then if ^ = be the linear equation or one of the linear equations of the system, the resultant will be of the form 0i^..., where <^i, ^, &c. are what the function <}> becomes upon substituting therein the different sets (a?i, yi, -2^1 ...)> (^i ya* -^^a**-) of the remaining (n — 1) equations -^ = 0, X ~ ^' ^* > comparing such expression with the given value of the resultant, we have expressed in terms of the coefficients of the functions y^, x* ^j certain symmetric functions which may be caHed the fundamental symmetric functions of the roots of the system -^ = 0, % = 0, &c. ; these are in fact the symmetric functions of the first degree in respect to each set of roots. By the aid of these fundamental symmetric functions, the other symmetric functions of the roots of the system -^ = 0, x ~ ^» ^- ^^7 ^ expressed in terms of the coefficients, and then combining with these equations a non-linear equation * = 0, the resultant of the system 4> = 0, ^^ = 0, x^^> ^- ^^^ ^ what the function *i*i . . . becomes, upon substituting therein for the different symmetric functions of the roots of the system •^ = 0, x = 0, &c. the expressions for these functions in terms of the coefficients. We thus pass from the resultant of a system ^ = 0, '^ = 0, x = 0, &c., to that of a system 4> = 0, V^ = 0, X " ^> ^» ^^ which the linear function <f> is replaced by the non-linear function *. By what haa preceded, the symmetric functions of the roots of a system of (n — 1) equations depend on the resultant of the system obtained by combining the (n—l) equations with an arbitrary linear equation ; and moreover, the resultant of any system of ri equations depends ultimately upon the resultant of a system of the same number of equations, all except one being linear; but in this case the linear equations determine the ratios of the variables or (disregarding a common factor) the values of the variables, and by substituting these values in the remaining equation we have the resultant of the system. The process leads, therefore, to the expressions for the synmietric functions of the roots of any system of (w— 1) equations, and also to the expression for the resultant of any system of n equations. Professor Schlafli discusses in the general case the problem of showing how the expressions for the fundamental symmetric functions lead to those of the other symmetric functions, but it is not necessary to speak further of this portion of his investigations. The object of the present Memoir is to apply the process to two particular cases, viz. I propose to obtain thereby the expressions for the simplest symmetric functions (after the funda- mental ones) of the following systems of two ternary equations; that is, first, a linear equation and a quadric equation ; and secondly, a linear equation and a cubic equation. First, consider the two equations («, 6, c, /, g, h^x, y, zy = 0, (a, /8, yj^x, y, z) = 0, and join to these the arbitrary linear equation (f , V> ?$^, y, ^) = 0, 456 ON THE SYMMETRIC FUNCTIONS OF THE ROOTS [149 then the two linear equations give and substituting in the quadratic equation, we have for the resultant of the three equations, (a, 6, c,/ g, hJi^^^ryr,, 7? - «?, a^-/9f)^ = 0, which may be represtented by (a, b, c, f, g, h$f 17, f)> = 0, where the coefficients are given by means of the Table. a b c f -2)8y 9, h a=> + >* + fi' 1 (a b = + y' + a' -2ya (-J*) c = + /3» + o» -a* + afi (D f = -Py 2(i»0 g = -ya + a)8 -P" + ;3y 2«0 h= -aP + ya ^Py -y" 2(^) viz. a = 67* + C)8* - 2/597, &c. But if the roots of the given system are then the resultant of the three equations will be and comparing the two expressions, we have a = 'X1X2 i b = yiy« » c = -2^1-8^2 i 2f = yi^2 + y^i> 2g = ZlX^ ^-z^i. 2h = ^iVi + arjyi, which are the expressions for the six fundamental symmetric functions, or symmetric functions of the first degree in each set, of the roots of the given system. By forming the powers and products of the second order a', ab, &c., we obtain linear relations between the symmetric fimctions of the second degree in respect to each set of roots. The number of equations is precisely equal to that of the 149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 457 symmetric functions of the form in question, and the solution of the linear equations gives — a* = x^x^, b* = yi V. he = ViZ^^t , ca = ZiXiZ^ , ab = x^iXttf,, 4f» 4g» 4h» 2bc = 2ca = 2ab = 2af = 2bg = 2ch = 4gh - 2af = a^»y^j + aJj^yiiTi, 4hf - 2bg = yi«-e:jic, + yj^iTja?!, 4fg - 2ch = -gTi^j?,^, + z^Xjy^ , 2bf = yi»y^, + ya^yi-gr^ , 2cg = V^,a?a + 2rj9iria?i , 2ah = iCi'^j + a;i*a?iyi, 2cf = Vya^i + ^a^^i , 2ag = Xi^^ + x^ZiX^ , 2bh = yi«aj,y8 + ya«a7iyi. Proceeding next to the powers and products of the third order a', a'b, &c., the total number of linear relations between the symmetric functions of the third degree in respect to each set of roots exceeds by unity the number of the symmetric functions of the form in question; in fact the expressions for abc, aP, bg*, ch', fgh, contain, not five, but only four symmetric functions of the roots; for we have abc = x^^Zy^ . x^iZ^y 4af» = {xjy^x^i + x^^x^z^) + ^lyiZiX^^^, 4bg» = (j/iZih/^^^ + j/j^aVi^O + ix^y^z^x^^^, 4ch* = {z^x^z^^ + z^cfzTjjf) + ^i^xZ^x^^^, 8fgh = {x^^x^i + x^}x^z^ ) ' + {yxZxV^i + y%z^xx^) - + 2a^yi-^i^a^j, + {Z^X^Z^} + Z^^ZtJI^ ) , C. II. 58 458 ON THE SYMMETBIO FUNCTIONS OF THB ROOTS [149 and consequently the quantities a, b, c, f, g, h, are not independent, but are connected by the equation abc - aP - bg« - ch« + 2fgh = 0, an equation, which is in fact verified by the foregoing values of a, &c. in terms of the coeflScients of the given sjrstem. The expressions for the symmetric functions of the third degree considered as Hinctions of a, b, c, f, g, h, are consequently not absolutely determinate, but they may be modified by the addition of the term \ (abc — af* — bg* — ch' + 2fgh), where \ is an indeterminate numerical coefficient. The simplest expressions are those obtained by disregarding the preceding equation for fgh, and the entire sjrstem then becomes : at ^_ /M S/ft 3 ^— U/i U/] , b' =»yiV» b«c = yi'^iy,"^, , c^a ^ JB| SC\Z^ cc^ f bc» = VxZ^^f , ca" ^ZiX^z^, ab« = xijf^^x^^, abc s= ohyiZiX^tSt^ 2a»f = x^^yiz^^ + x^y^xO^. 2b«g= y^z^x^}-\-y}z^^^, 2c»h = Zx^x^^^ + z^x^iZx\ 2a fi ^~ x^ z^fjc^ "T x^ i9]M>| , 2b«h = yx^x^^^ + y2Si>{yx\ 2c»f = Vy^,« + z^y,z,\ 2a'h = x^z}x^ + x^z^Xy , 2b«f = yi»a:,«yj + y,»a^«yi> 2c*g = z^%^z^ + z^h/^^Zx , 2bcf = yx^z^^f + y}z^xz^ , 2cag= z^XiZ^ +z^XiZiX^, 2abh= x^ix^^ ^ x^^iOxyx\ 2bcg = yxz^x^^^ + y^ix^xZx , 2cah = ZiX^x^^2 + z^x^iZ^ , 2abf = x^j^x^^t + x^^xiy^z^ , 149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 459 2bch= VxZ^^H^t + yfz^^xZx , 2caf = z^x^x^^^ + zix^yyz^ , 2abg= x^%x^^^ + iJc^y^iyxZx. 4af» -2abc = a?iy,V^j+a?j^aV^, 4bg» -2abc = yiV^aVs + yV^q^y,, 4ch« -2abc = Zx^y}Z% + Z^}\)^Z^ , 4bf» -2b*c = yxy^%-^yiyxz^. 4cg» -2c«a = z^z^i + z}z^x^ , 4a.h« -2a«b = x^x^^ + x}x^^. 4cP -2bc« = Zx^%z^ + z^yx^z^ , 4ag». -2ca« = x^z^x^ + a^ V^ » 4bh«. -2ab» = yx*^9%-^yMY> 4agh- -2a2f = x^^x^^^ + x^x^y^z^. 4bhf- - 2b«g = yi*x^^^ -h yj'x^iZi, 4cfg - -2c«h = Z^^^% + ZfXTjj^Zy^ , 4bgh' - 2abf = yxZx^%y%-^ y^z^y^. 4chf - 2bcg = z^x^iz^ + z}x^^z^ , 4afg > -2cah = O^xZix^-^ O^^^^x. 4cgh • -2acf = yxz^xfz^ -h y^ix^Zx , 4ahf -2bag = ZxX^}x^-\- z^^y^x^. 4bfg - 2cbh = i^xUxZ^y^^r x^fz^y^. SPg- - 4chf . -2bcg = z^x^} + z^m^. 8g«h. -4a% - -2cah = x^y^} + xiy^z^. 8h«f- - 4bgh • - 2abf = yxz^i + y^z^x^. 8fg»- -4chg -2acf = V^'ya + V^'yi, 8gh«. -4afh- - 2bag = a^yfz^^- x^y^z^, 8hf«. -4bgf . - 2cbh = y^zix^-^-yiz^Xx, 8f» ■ - 6bcf = yxzi + y,V, 8g» -6cag = z^x{ + z}x^y 8h» . - 6abh = «i V + «a V- Secondly, consider the system of equations (a, 6, c, /, flf, A, i, j, fc, fja?, y, ^)» = 0, (a, /9, 7]^a?, y, -^) = 0, 58—2 460 ON THE SYMMETRIC FUNCTIONS OF THE ROOTS [149 where the cubic function written at full length is Joining to the system the linear equation (f I % ?$«. y, ^) = 0, the linear equations give and the resultant is which may be represented by (a, b, c, f, g. h, i, j, k, l$f , 17, ?)> = 0, where the coefficients a, b, &c. are given by means of the Table : — a f a = b = c = -3-ya» -3ai8» + 3i8V + 87*0 + 3a')3 -2o/3y f f = g = b = -/a + /3V -a* -laJfiy -2a)3y + 2ya' + 20/3' + 2)8y' -2o»/3 -2y*a 3« 3f«i, • 1 = • J = k= -A -o»6 + a» + 2a^y + 2o/8y + 2aj8y -a/3» + ya' -ya« + /8y' 3{^ 1 3^ ■ 1 = -y«« -«y3» -H + a»/3 +y3'y + 7*0 6M; VIZ. a = 67* - ciS* - 3/i87» + 3ti8«7, &c. But if the roots of the given system are then the resultant of the three equations may also be represented by and comparing with the former expression, we find : a ^^ X\X^-^y b = yiy^a^j, C s= Z\Z^^^ 149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 461 3f = yiy^t + y^^x + y^iz^. 3g = ZiZ^x^ + z^z^ + z^iXi, 3i = y^z^t + ya«r^i + y^^z^, 3j = -e^iiTa^s + Z^X>^ + ZgXiX^y 3k = ay^^, + iC8yayi+ a?,yiyj, But there is in the present case a relation independent of the quantities a, &c., viz. we have (a, )8, 7$iCi, yu ^i) = 0, (a, )8, y^x^, y,. -8^,) = 0, (a, $, 7$j^, y„ z,) = 0, and thence eliminating the coefficients (a, /8, 7), we find By forming the powers and products of the second degree a', ab, Ac, we obtain 55 equations between the symmetric functions of the second degree in each set of roots. But we have V = = a symmetric ftmction of the roots, and thus the entire number of linear relations is 56, and this is in fact the number of the synmietric functions of the second degree in each set. I use for shortness the sign S to denote the sum of the distinct terms obtained by permuting the different sets of roots, so that the equations for the fundamental symmetric functions are — a = ^^2^> b= y,y^„ C = ZiZ^Zit 3f = S yij/j^s, 3g=S-?i^2a?8, Sh = SxjX^i, 3i =SyiZ^z^, 3j =SziX^3, 3k = S x^^t, 61 =Sx{y^^\ then the complete system of expressions for the symmetric functions of the second order is as follows, viz. a« = x^x^x^, b» = yWyz\ c« = z^^z^H,\ be = yiz^y^^^t, ca -^ Zi^XiZ^flc^ZfjX^f ab= Xiy^x^^^t, 462 ON THE SYMMBTBIC FUNCTIONS OF THE BOOTS [l49 3af =^ S XitfiX^^^Xt, 3ch = S siiXiZ^x^^z^, 3bf =iSyi»y,»y,^„ 3cg=iS-fxV'8^s«8, 3ah = Sx^x^x^i, 3cf = S yiz,y^ztz^\ Ssig =^ S z,x,z,fc^\ Sbh = S x^ix^^^\ 3ai - S x^^z^^z^j^, 3bj ^Sj/iz^x^^^^, 3ck = S ZiXiy^^iZs, 3bi = 8 yi^^^tz^, 3cj ^Szi^z^z^, 3ak = jS ari«a?3^jfl^„ 3ci ^SyiZiZ^z^, 3aj =SziXiXi^Xi\ ahk = Sx^^^y^\ 6al =Sxi^x^iZ^^, 6bl = iS yi*y^,a^„ 6cl = S Zi^z^x^^z^, 9f« - 6bi = S y,«y,V, 9g» -6cj =Sz,W^\ 9h» - 6ak = iS Xx"a:i«y,«, 9i'» - 6cf = iS yl«-^a«-^,^ 9j« -6ag = S^,»a:,%», 9k« - 6bh = S a?,«y,V. 9fg - 3ck = iS Xjy^^^s\ 9gh-3ai ^Sy^z^z^^^\ 9hf-3bj ^SziXjX^^i^ 149] OP CERTAIN SYSTEMS OF TWO EQUATIONS. 463 9jk - 3af = Sxi^x^^j^i, 9ki - 3bg = fif y^y^^z^^, 9ij — 3ch = S z^z^xi^^^, 9f i - 3bc = S y,^y,z^,\ 9gj — 8ca = S z^z^^x^y 9hk~ Sab = fif x^^x^^^, 3 ( Q + gk + hi - 1«) = Sx^,z^x^^^, 3(2fj -gk-hi + l«) = fif^iy,a?4^^,«, 3(2gk -hi -fj +\*) = 8y,z,y,z^,\ 3(2hi -fj -gk + l«) = flf^ia?,^^,y,«, 3(6fl -3ki-.bg) =flfay^^,V, 3(6gl -3ij-ch) =8y,z,z,W, 3(6hl -3jk-af) =^Sz^x^x^^^\ 3(6il -3fg-ck) =SziX^^%\ 3(6jl -3gh-ai) ^Sx^,zW. 3(6kl-3hf-bj) ^Sy,z,x,%\ 6(~fj-gk-hi + 41«) = fifa:i»y,V. As an instance of the application of the formulae, let it be required to eliminate the variables from the three equations, (a, 6, c, /, g, h, i, j, A, i$a;, y, ^)» = 0, (a\ b\ c\ /, ^, K Jx, y, zf = 0, (a, yS, 7 $a;, y, 2^) = 0. This may be done in two different ways; first, representing the roots of the linear equation and the quadric equation by (a?i, yi, ^,), (a^, y,, z^, the resultant will be (tt, ...$071, yi, ^,)».(a,...Ja;i, y,, -erj)*, which is equal to a* aJj'^Tj' + &C., where the symmetric functions x^x^, &c. are given by the formulae a'* = ajj'a?,*, &c., in which, since the coefficients of the quadratic equation are {a\ h\ c\ f\ gf^ h'\ I have written a' instead of a. Next, if the roots of the linear equation and the cubic equation are represented by {xi, yi, z^, (a?,, yj, z^, (x^, y,, -?,), then the resultant will be (tt', ...$da, yi. ^,)'.(a',...$d:2, ya, z^y(a\ ...^x^, y„ -^j^, 464 ON THE SYMMETRIC FUNCTIONS OF THE ROOTS &C. [149 which is equal to a'* ayfxfx^ + &c., the symmetric functions x^^x^x^, &a being given by the formulaB d?=^x^x^x^, &c. The expression for the Resultant is in each case of the right degree, viz. of the degrees 6, 3, 2, in the coefficients of the linear, the quadric, and the cubic equations respec- tively: the two expressions, therefore, can only differ by a numerical factor, which might be determined without difficulty. The third expression for the resultant, viz. (where (a?,, yi, Zi\,..(x^, y^, z^ are the roots of the cubic and quadratic equations) compared with the foregoing value, leads to expressions for the fundamental sjrmmetric functions of the cubic and quadratic equations, and thence to expressions for the other symmetric functions of these two equations; but it would be difficult to obtain the actually developed values even of the fundamental symmetric functiona I hope to return to the subject, and consider in a general point of view the question of the formation of the expressions for the other symmetric functions by means of the ex- pressions for the fundamental symmetric functiona 150] 465 150, A MEMOIR ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN SYSTEMS OF EQUALITIES AMONG THE ROOTS OF AN EQUATION. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 727 — 731. Received December 18, 1856, — Read January 8, 1857.] It is well known that there is a symmetric function of the roots of an equation, viz. the product of the squares of the differences of the roots, which vanishes when any two roots are put equal to each other, and that consequently such function expressed in terms of the coefficients and equated to zero, gives the condition for the existence of a pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of his earlier papers in the Philosophical Magazine, that the like method could be applied to finding the conditions for the existence of other systems of equalities among the roots, viz. that it was possible to form symmetric functions, each of them a sum of terms containing the product of a certain number of the differences of the roots, and such that the entire function might vanish for the particular system of equalities in question ; and that such functions expressed in terms of the coefficients and equated to zero would give the required conditions. The object of the present memoir is to extend this theory and render it exhaustive, by showing how to form a series of types of all the different functions which vanish for one or more systems of equalities among the roots; and in particular to obtain by the method distinctive conditions for all the different systems of equalities between the roots of a quartic or a quintic equation, viz. for each system con- ditions which are satisfied for the particular system, and are not satisfied for any other systems, except, of course, the more special systems included in the particular system. The question of finding the conditions for any particular system of equalities is essen- tially an indeterminate one, for given any set of functions which vanish, a function syzygetically connected with these will also vanish; the discussion of the nature of the C. II. 59 466 ON THE CONDITIONS FOR THE EXISTENCE OP GIVEN [150 syzygetic relations between the diflFerent functions which vanish for any particular system of equalities, and of the order of the system composed of the several conditions for the particular system of equalities, does not enter into the plan of the present memoir. I have referred here to the indeterminateness of the question for the sake of the remark that I have availed myself thereof, to express by means of invariants or covariants the different systems of conditions obtained in the sequel of the memoir; the expressions of the different invariants and covariants referred to are given in my 'Second Memoir upon Quantics,' Philosophical TranscLCtions, vol. CXLVI. (1856), [141]. 1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call the roots a, 13, % S, e. Write 12 = 2<^(a-i8)Ml2.13 = 2<^(a-i8y(a-7)"», 12.34 = 2^ (a — i8)'(7 — 8)**, &c., where ^ is an arbitrary function and i, m, &a are positive integers. It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 and 23.25, &c., denote identically the same sums. Two types, such as 12.13 and 14.15.23.24.25.34.35.45, may be said to be complementary to each other. A par- ticular product (a — yS)(7 — 8) does or does not enter as a term (or factor of a term) in one of the above-mentioned sums, according as the type 12.34 of the product, or some similar type, does or does not form part of the type of the sum; for instance, the product (a — i8)(7 — S) is a term (or factor of a term) of each of the sums 12.34, 13.45.24, &c., but not of the sums 12.13.14.15, &a 2. If, now, we establish any equalities between the roots, e.g. ci = l3, 7 = 8, the effect will be to reduce certain of the sums to zero, and it is easy to find in what cases this happens. The sum will vanish if each term contains one or both of the &ctors ^"ffi 7— Si i.e. if there is no term the complementary of which contains the product (a — yS) (7 — 8), or what is the same thing, whenever the complementary type does not con- tain as part of it, a type such as 12.34. Thus for the sum 14.15.24.25.34.35.45, the complementary type is 12.13.23, which does not contain any type such as 12.34, i. e. the sum 14.15.24.25.34.35.45 vanishes for a = y3, 7 = 8. It is of course clear that it also vanishes for a = yS = e, 7 = 8 or a = yS = 7 = 8, &a, which are included in ^ = fi» 7=8. But the like reasoning shows, and it is important to notice, that the sum in question does not vanish for a = ^ = y: and of course it does not vanish for a = ^. Hence the vanishing of the sum 14.15.24.25.34.35.45 is characteristic of the system a = yS, 7=8. A system of roots a, yS, 7, 8, e may be denoted by 11111; but if a = y3, then the system may be denoted by 2111, or if a = i8, 7 = 8, by 221, and so on. We may then say that the sum 14.15.24.25.34.35.45 does not vanish for 2111, vanishes for 221, does not vanish for 311, vanishes for 32, 41, 5. 3. For the purpose of obtaining the entire system of results it is only necessary to form Tables, such as the annexed Tables, the meaning of which is sufficiently explained by what precedes: the mark (x) set against a type denotes that the sum represented by the complementary type vanishes, the mark (o) that the complementary type does not vanish, for the system of roots denoted by the symbol at the top or bottom of the column; the complementary type is given in the same horizontal line with the original type. It will be noticed that the right-hand columns do not extend to the foot of the Table ; the reason of this of course is, to avoid a repetition of the same type. Some of BTSTEH8 OF EQUAUTIES AUONQ THE ROOTS OF AN EQUATION. 467 150] the types at the foot of the Tables are complemeDtaiy to themselves, but I have, not- withstauding this, given the complementary type in the form under which it naturally presents itself. 4. The Tahles are: Table for the equal Boots of a Quartic. 211 22 31 4 14 . 23 . 24 . 34 o o ■s' o 14 . 23 . S4 . 34 o o o X 14 . 23 . 24 . 34 o o o » 13 . 14 . 23 . 24 a o X X "IT 23 . 24 . 34 — — — 84 14 . 23 . 34 sn 22 31 4 23 14 . 24 . 34 Table for the equal Boots of a Quintia Sill 221 311 32 *1 fi ~^ T 7 o X X X X X o X X X X X o o X X X ly o X x X X ix o o X X xlx o X X X jx o y. o y X |x o X X X X X o o a X X > o o X X X X o X X X X O X X X X o X X o o o X X X o o X X X o o o X X X o o X X X o a o o X X o o X X X ^ Bill ssT 311 32 41 7 2111 2« 51 32 41 b 35 . 46 o ~~^ ~^ a 35 . 45 a X 35 . 45 o X 35 . 45 o a X X 36 . 46 o o a y 34 . 35 o o X X 35 . 46 o o X X 35 . 46 o o X X X 35 . 45 o o X X 36 . 46 o o X X X 36 . 46 o X X X 36 . 46 o o X X 34 . 45 o o X X 34 . 36 o o X X X X 35 . 45 35 . 45 2111 221 311 32 41 b 36 . 45 ' 35 . 46 34 . 36 26 . 36 The two Tables enable the discussion of the theory of the equal roots of a quartic or quintic equation: first for the quartic: 5. In order that a quartic may have a pair of equal roots, or what is the same thing, that the system of roots may be of the form 211, the type to be considered is 12.13.14.23.24.34; 59—2 468 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN [l50 this of course gives as the function to be equated to zero, the discriminant of the quartic. 6. In order that there may be two pairs of equal roots, or that the system may be of the form 22, the simplest type to be considered is 14.24.34; this gives the function which being a covariant of the degree 3 in the coefficients and the degree 6 in the variables, can only be the cubicovariant of the quartic. 7. In order that the quartic may have three equal roots, or that the system of roots may be of the form 31, we may consider the type 13.14.23.24, and we obtain thence the two functions 2(a-7)(a-S)(iS-7)(^-S), 2(a-7)«(a-8)(^-7)(/3-8)', which being respectively invariants of the degrees 2 and 3, are of course the quadrin- variant and the cubinvariant of the quartic. If we had considered the apparently more simple type 12.34, this gives the function which is the quadrivariant, but the cubinvariant is not included under the type in question. 8. Finally, if the roots are all equal, or the system of roots is of the form 4, then the simplest type is 12; and this gives the function a covariant of the degree 2 in the coefficients and the degree 4 in the variables ; this is of course the Hessian of the quartic. Considering next the case of the quintic: 9. In order that a quintic may have a pair of equal roots, or what is the same thing, that the system of roots may be of the form 2111, the type to be considered is 12.13.14.15.23.24.25.34.35.45; this of course gives as the function to be equated to zero, the discriminant of the quintic. 150] SYSTEMS OP EQUALITIES AMONG THE ROOTS OF EQUATION. 469 10. In order that the quintic may have two pairs of equal roots, or that the system of roots may be 221, the simplest type to be considered is 14.15.24.26.34.35.45; a type which gives the function 2 (a - S)(a - e)(^ - 8)(yS - e)(7 - S)(7 - e)(B - €)« (a; - ay)' (x - ySy)« (x - 7y)». This is a covariant of the degree 5 in the coefiBcients and of the degree 9 in the variables ; but it appears from the memoir above referred to, that there is not any irreducible covariant of the form in question; such covariant must be a sum of the products (No. 13)(No. 20), (No. 13)(No. 14)», (No. 15)(No. 16) (the numbers refer to the Cova- riant Tables given in the memoir), each multiplied by a merely numerical coefficient. These numerical coefficients may be determined by the consideration that there being two pairs of equal roots, we may by a linear transformation make these roots 0, 0, oo , oo , or what is the same thing, we may write a = 6 = e=/=0, the covariant must then vanish identically. The coefficients are thus found to be 1, — 4, 50, and we have for a covariant vanishing in the case of two pairs of equal roots, 1 (No. 13)(No. 20) - 4 (No. 13)(No. Uy + 50 (No. 15)(No. 16) [or in the new notation AH — ^AB* + 50CD]. In fact, writing a = 6 = e=/=0, and rejecting, where it occurs, a factor ic'y*, the several covariants become functions of ex, di/; and putting, for shortness, x, y instead of ex, dy, the equation to be verified is 1 . 10(a: + y)(6a:* + Sa^y + 28«y + Sxy^ + 6y*) - 4i.lO{x'\-y){Za? + %xy + Zr/'y + 50(6a;* + &ry + 6y»)(a^ + ic'y + a?y« + y») = ; and dividing out by {x + y) and reducing, the equation is at once seen to be identically true. 11. In order that the quintic may have three equal roots, or that the system of roots may be of the form 311, the simplest type to be considered is 12.13.23.45; this gives the function 2(a-yS)>(i8«7)'(7-«)'(S-«)*, which being an invariant, and being of the fourth degree in the coefficients, must be the quartinvariant of the quintic [that is No. 19, = 0\ The same type gives also the function 2(a-^)«(i8-7)«(7-a)«(S-.6)«(a:-.Sy)»(a:-.6y)», 470 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN SYSTEMS, &C. [150 which is a covariant of the degree 4 in the coefficients and the degree 4 in the variables; and it must vanish when a = 6 = c = 0, this can only be the covariant 3 (No. 20) -2 (No. 14)>, [=3ir-25«], which it is clear vanishes as required. 12. In order that the quintic may have three equal roots and two equal roots, or that the system of roots may be of the form 32, the simplest type to be con- sidered is 12.13.14.15, which gives the function 2(a-^)(a-7)(a-S)(a-e)(a;-ySy)«(^-7y)*(a?-Sy)«(^-6y)«, a covariant of the degree 4 in the coefficients, and the degree 12 in the variables; and it must vanish when a = 6 = c = 0, e=/=0; this can only be the covariant 3 (No. 13)»(No. 14) -25 (No. 15)», [= 3^»B - 25C*], which it is clear vanishes as required. 13. In order that the quintic may have four equal roots, or that the system may be of the form 41, the simplest type to be considered is 12.34, which gives the function 2(a-yS)«(7-S)'(^-6y)», a covariant of the degree 2 in the coefficients, and of the same degree in the variables; this can only be the covariant (No. 14), [=-B]. 14. Finally, in order that all the roots may be equal, or that the system of roots may be of the form 5, the type to be considered is 12; and this gives the function a covariant of the degree 2 in the coefficients, and the degree 6 in the variables, and this can only be the Hessian (No. 15), [=s C]. It will be observed that all the preceding conditions are distinctive; for instance, the covariant which vanishes when the system of roots is of the form 311, does not vanish when the system is of the form 221, or of any other form not included in the form 311. 151] 471 151. TABLES OF THE STUEMIAN FUNCTIONS FOR EQUATIONS OF THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES. [From the Philosophical Transdctiona of the Royal Society of London, vol. CXLVII. for the year 1857, pp. 733—736. Received December 18, 1856,— Read January 8, 1857.] The general expressions for the Sturmian functions in the form of determinants are at once deducible firom the researches of Professor Sylvester in his early papers on the subject in the Philosophical Magazine, and in giving these expressions in the Memoir 'Nouvelles Recherches sur les Fonctions de M. Sturm,' Liouville, t. xiii. p. 269 (1848), [65], I was wrong in claiming for them any novelty. The expressions in the last-mentioned memoir admit of a modification by which their form is rendered some- what more elegant; I propose on the present occasion merely to give this modified form of the general expression, and to give the developed expressions of the functions in question for equations of the degrees two, three, foUr, and five. Consider in general the equation Cr = (a, 6, ... j, k^x, 1)», and write P = (a, 6, ... jJix, ir'\ Q=(6, ... j^fcJix, l)--\ then supposing as usual that the first coefficient a is positive, and taking for shortness th •"" 1 w """ 1 fh "■" 2 ?ij, n,, &c. to represent the binomial coefficients — = — , :.—^ , &c. corresponding to the index (n — 1), the Sturmian functions, each with its proper sign, are as follows, viz. 472 TABLES OF THE STURMIAN FUNCTIOKS FOR ri5i U. P. P. Q a , b - *P. P. 'a. • , iij6, a , *«, Q 6, . fliC^ b i I + a^P, xP, P. a»e. xQ. Q . &c a. I a, njb. a, ry:. 6. 6. . w,c, 6 where the terms contaimng the powers of x, which exceed the degrees of the several functions respectively, vanish identically (as is in fact obvious from the form of the expressions), but these terms may of course be omitted ab initio. The following are the results which I have obtained; it is well known that the last or constant function is in each case equal to the discriminant, and as the expressions for the discriminant of equations of the fourth and fifth degrees are given, Tables No. 12 and No. 26 [Q', see 143] in my 'Second Memoir upon Quantics'(0. I have thought it sufficient to refer to these values without repeating them at length. Table for the degree 2. The Sturmian functions for the quadric (a, 6, c$a?, 1)* are c + 1 \l^x, ly. (! a+ I 6 + 1 5*. 1). ac—\ 6» + l Table for the degree 3. The Sturmian functions for the cubic (a, 6, c, d$a?, ly are a+l b + 3 e + 3 (i+1 5*^ 1)*. 1 Philotophieal Trantactions, t. cxlvi. p. 101 (1856), [141]. 151] EQUATIONS OF THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES. 473 a + 1 6 + 2 c+l '$.«>, 1)', $«. 1). 1 a'cP + 1 abed + 6 ' ac" -4 ' bd" -4 b'c^ -3 Table for the degree 4. The Sturmian functions for the quartic (a, 6, c, d, e$x, 1\ are ( a+ 1 6 + 4 c + 6 (^ + 4 6 + 1 5*. 1)*' a+ 1 6 + 3 c + 3 c^+l $*. 1)'. ac— 3 6^+3 ac;-3 6c +3 ae-\ bd-^l Ja^. 1)'. 3( a^ce - 1 a^cfe + 1 a»c;» + 3 a6cc —4 ah^e + 1 ahd:" -1 o6c€? — 14 ac^d +3 ac» + 9 6»e +3 6»c^ + 8 6»cc; -2 6V - 6 $«^ 1), aV+1 Disct. Tab. No. 12. Table for the degree 5. The Sturmian functions for the quintic (a, 6, c, d, e, /$a:, 1)" are a + 1 6 + 5 c+10 ef+10 e + 5 /+i r^x, i)», C. II. 60 474 TABLES OF THE 8TUEM1AN FUNCTIONS, &C. [151 a + 1 6 + 4 c + 6 d-h4 e+l Jix, ly, ( oc - 4 6« +4 arf-6 6c +6 a« — 4 bd+i a/-l 6« +1 "ga:, 1)', 2( a«ce - 8 a V - 2 a»c(/* + 3 a>^ +18 a«^ +12 a6c/- 11 a6»c + a a6y+ 2 abde-- 3 a6crf- 76 abce - 42 ac"e + 8 ac» +48 a6rf« -12 6y + 8 b^d +40 cuM +32 6«cc - 5 6V -30 6»e + 30 b^cd - 20 \', 1)", 2( aV" - 2 a^dp + 3 a^def + 24 aV/ - 8 aV - 32 a^bcP - 11 a>6y> + 2 a^bdef + 58 a»Me* + 264 a»6c» + 8 a'ftcc/" — 52 aV«/* + 104 a^bd}/-- 96 a^ccP/ - 156 a^(?df + 64 a«ccfe« - 96 aVtf> + 352 a*cPe + 108 a'ccPe - 938 a6y* +• 8 a»(£* + 432 a6*c«/ - 266 aJt^ef + 28 ab^d^f - 8 oftW - 970 ah^d^ + 35 o^'cPe + 120 abi^df -^ 584 abi^de + 2480 abi?^ + 120 aJl^cdf -^ 264 a6cc?e - 360 ahcd^ - 1440 oc*/ - 288 a6cy - 192 W?dA + 160 ac*« - 960 6V + 120 cu^d^ + 640 6»ai/ - 320 6*(^ - 160 6»ce« - 75 6V + 450 6»(?6 + 200 l^cde - 1400 6»cy + 180 6»d» + 800 6«c»^ - 100 6V/ + 120 6V6 + 600 6Vrf» - 400 aV* + l + Ac, Disct Tab. No. 26, [Q']. 5^, 1), 152] 475 152, A MEMOIR ON THE THEORY OF MATRICES. [From the Philosophical Transactions of the Royal Society of London, vol. CXLVili. for the year, 1858, pp. 17 — 37. Received December 10, 1857, — Read January 14, 1858.] The term matrix might be used in a more general sense, but in the present memoir I consider only square and rectangular matrices, and the term matrix used without qualification is to be understood as meaning a square matrix ; in this restricted sense, a set of quantities arranged in the form of a square, e.g. ( a , b , c ) a' , V , d a", V\ d' is said to be a matrix. The notion of such a matrix arises naturally firom an abbreviated notation for a set of linear equations, viz. the equations z = ax + 6y '\-cz , Y^a'x + Vy + dz , Z ^(j^'x-^V'y^c'z, may be more simply represented by (Z, F, ^ = ( a , 6 , c '^x, y, «), a' , V , d a", V\ d' and the consideration of such a system of equations leads to most of the fundamental notions in the theory of matrices. It will be seen that matrices (attending only to those of the same order) comport themselves as single quantities; they may be added, 60—2 476 A MEMOIB ON THE THEORY OF MATRICES. [152 multiplied or compounded together, &c. : the law of the addition of matrices is pre- cisely similar to that for the addition of ordinar}' algebraical quantities; as regards their multiplication (or composition), there is the peculiarity that matrices are not in general convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function, of a matrix. I obtain the remarkable theorem that any matrix whatever satisfies an algebraical equation of its own order, the coeflBcient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefiBcient being in bet the determinant; the rule for the formation of this equation may be stated in the following condensed form, which will be intelligible after a perusal of the memoir, viz. the determinant, formed out of the matrix diminished by the matrix considered as a single quantity involving the matrix unity, will be equal to zero. The theorem shows that every rational and integral function (or indeed every rational function) of a matrix may be considered as a rational and integral function, the degree of which is at most equal to that of the matrix, less unity; it even shows that in a sense, the same is true with respect to any algebraical function whatever of a matrix. One of the applications of the theorem is the finding of the general expression of the matrices which are convertible with a given matrix. The theory of rectangular matrices appears much less important than that of square matrices, and I have not entered into it further than by showing how some of the notions applicable to these may be extended to rectangular matrices. 1. For conciseness, the matrices written down at full length will in general be of the order 3, but it is to be understood that the definitions, reasonings, and con- clusions apply to matrices of any degree whatever. And when two or more matrices are spoken of in connexion with each other, it is always implied (unless the contrary is expressed) that the matrices are of the same order. 2. The notation ( a , 6 , c ^x, y, z) a', 6', c' represents the set of linear functions ((a, 6, c\x, y, z\ (a', h\ d\x, y, z\ (a", 6", c"\x, y, z)\ so that calling these (X, F, Z), we have (X, F, ^ = ( a , 6 , c \x, y, z) and, as remarked above, this formula leads to most of the fundamental notions in the theory. 152] A MEMOIR ON THE THEORY OF MATRICES. 477 3. The quantities {X, Y, Z) will be identically zero, if all the terms of the matrix are zero, and we may say that ( 0, 0, .) 0, 0, 0, 0, is the matrix zero. Again, {X, T, Z) will be identically equal to {x, y, z), if the matrix is (1. 0. ) 0, 1. ! 0, 0, 1 : and this is said to be the matrix unity. We may of course, when for distinctness it is required, say, the matrix zero, or (as the case may be) the matrix unity of such an order. The matrix zero may for the most part be represented simply by 0, and the matrix unity by 1. 4. The equations (Z, F, Z) = { a, b, c Jix.y. z\ X\ F, Z') = ( « , /S , 7 $.^, y, z) a' , 6' , c' ^" J." V a , , c «'. /3', 7' a", /8", 7" give {X + X, Y+T, Z + Z) = { a +a , b +0 , c +y Jix, y, z) a' +a', V +yS', c'+7' a" + a", 6" + ^", (Z' + y and this leads to (o+a, b +P , c+7 ) = (a, b , c ) + (o, fi , 7 ) o' +0', b' +ff, c' +7' o" + a", 6"+/3", c"+7" a' , b' , c' a", b", c" as a rule for the addition of matrices ; that for their subtraction is of course similar to it. 5. A matrix is not altered by the addition or subtraction of the matrix zero, that is, we have M ±0 = M. The equation L==M, which expresses that the matrices Z, M are equal, may also be written in the form Zcr-M^^O, Le. the difference of two equal matrices is the matrix zero. 6. The equation L^--My written in the form Z + Jf^O, expresses that the sum of the matrices L, M \a equal to the matrix zero, the matrices so related are said to be opposite to each other ; in other words, a matrix the terms of which are equal but oppo- site in sign to the terms of a given matrix, is said to be opposite to the given matrix. 478 A MEMOIR ON THE THEORY OF MATRICES. [152 7. It is clear that we have L -{-M = M + L, that is, the operation of addition is commutative, and moreover that (L + M) + JV = Z + (M -i-If^^L + M + N, that is, the operation of addition is also associative. 8. The equation written under the forms (X, Y, Z) — { a , 6 , c $ww?, my, mz) (X, T, Z) = m( a , 6 , c $a:, y, z) = ( via , mb , mc ^x, y, z) a\ 6' , c' a , , c ma' , mi/ , mc' ma", fii6", m&' gives m( a y 6, c ) = ( ma , mh , mc ) a\ h\ c' a'\ h'\ c" tna' , mV , mc' ?7ia", m6", mc" as the rule for the multiplication of a matrix by a single quantity. The multiplier m may be written either before or after the matrix, and the operation is therefore com- mutative. We have it is clear m{L-\-M) — mL + mJf, or the operation is distributive. 9. The matrices L and mL may be said to be similar to each other; in particular, if m = 1, they are equal, and if m = — 1, they are opposite. 10. We have, in particular, m ( 1, 0, ) = ( m, 0, ), 0, m, 0, 0, m 0, 1, 0, 0, 1 or replacing the matrix on the left-hand side by unity, we may write m=={m, 0, ); 0, m, 0, 0, m the matrix on the right-hand side is said to be the single quantity m considered as involving the matrix unity. 11. The equations (Z, F, Z) = ( a, a , 6 , c $a?, y, z\ (a?, y, ^) = ( a , ^ , 7 $f, 1;, {), 6', C a', /3', y i", c" a", /3", y 152] A MEMOIR ON THE THEORY OF MATRICES. 479 give (X,7.Z) = (A, B, C IJf , 17. ?) = ( o , 6, c ^a. A', B, C A", B", C" a'. V, a , 6 , jf a, fjt Q>l ^11 and thence, substituting for the matrix {A , B , C ) A', R. C I A", F', C" its value, we obtain {{a.b.c^a. a', a"), {a,b ,c^P, &, /8"). (a , 6 . c 57, 7. 7") ) - ( « . & . c $ a . ^ . 7 ) (a' .b'.c' $a, a', O, (a' . 6' , c' 3[y9, ff, /3"), (a' , 6' . c' $7, 7'. 7") (a", 6", c"$a, a', a"), (a". 6", c"$/8, ff, n. (o", 6". c"$7. 7'. 7") o',6',c' _// 1;// -^ «',/3'.7' a", /8". 7" as the rule for the multiplication or composition of two matrices. It is to be observed, that the operation is not a commutative one; the component matrices may be distinguished as the first or further component matrix, and the second or nearer component matrix, and the rule of composition is as follows, viz. any line of the com- pound matrix is obtained by combining the corresponding line of the first or further component matrix successively with the several columns of the second or nearer com- pound matrix. [We may conveniently write (a. «', a"). (A /9'. ^'). (7. 7'. 7") (a , b. c) if f> ij (a', b', c') t> it if (0". b". c") n » n to denote the left-hand side of the last preceding equation.] 12. A matrix compounded, either as first or second component matrix, with the matrix zero, gives the matrix zero. The case where any of the terms of the given matrix are infinite is of course excluded. 13. A matrix is not altered by its composition, either as first or second component matrix, with the matrix unity. It is compounded either as first or second component matrix, with the single quantity m considered as involving the matrix unity, by multiplication of all its terms by the quantity m:. this is in fact the before-mentioned rule for the multiplication of a matrix by a single quantity, which rule is thus seen to be a particular case of that for the multiplication of two matrices. 14. We may in like manner multiply or compound together three or more matrices: the order of arrangement of the factors is of course material, and we may 480 A MEMOIR ON THE THEORY OF MATRICES. [152 distinguish them as the first or furthest, second, third, &c., and last or nearest component matrices: any two consecutive factors may be compounded together and replaced by a single matrix, and so on until all the matrices are compound^ together, the result being independent of the particular mode in which the composition is effected; that is, we have L.MN^LM .N ^ LMN, LM.NP^L. MN.P, &a, or the operation of multiplication, although, as already remarked, not commutative, is associative. 15. We thus arrive at the notion of a positive and integer power 2> of a matrix L, and it is to be observed that the different powers of the same matrix are con- vertible. It is clear also that p and q being positive integers, we have L^.L^^L^^^, which is the theorem of indices for positive integer powers of a matrix. 16. The last-mentioned equation, 2>.Z« = 2>+«, assumed to be true for all values whatever of the indices p and q, leads to the notion of the powers of a matrix for any form whatever of the index. In particular, 2> . Z® = J!> or Z<* = 1, that is, the 0th power of a matrix is the matrix unity. And then putting p = l, } = — 1, or /> = — !, g=l, we have L . L"^ = Z~^ . Z = 1 ; that is, L~\ or as it may be termed the inverse or reciprocal matrix, is a matrix which, compounded either as first or second component matrix with the original matrix, gives the matrix unity. 17. We may arrive at the notion of the inverse or reciprocal matrix, directly from the equation (Z, F. Z) = ( a , 6 , c Jxy y, z\ a' , 6' , c' «" Uf V a , by c in fact this equation gives {XyyyZ) = (A, A\ ^"$Z, F, Z) = (( a, 6, c )-^$Jr, F, Z), 5, B, B' a , , c and we have, for the determination of the coefficients of the inverse matrix, the equations (A. A', ^"$o, 6, c ) = ( 1. 0, ), or reciprocal B, ff, B' C, C, C" a', V, c' a", b", c" 0. 1, 0, 0, 1 (a . b, c 11 A, A', A" ) = ( 1, 0, ), a' , 6' , c' a , 6", .// B, B, B' C, C, C" 0, 1, 0, 0. 1 s 152] A MEMOIR ON THE THEORY OF MATRICES. 481 which are equivalent to each other, and either of them is by itself sufficient for the complete determination of the inverse or reciprocal matrix. It is well known that if V denote the determinant, that is, if V = a , 6 , c a' , 6' , c' ^" J." «" a , , c then the terms of the inverse or reciprocal matrix are given by the equations V 1, 0, 0, b', c' 0, h". c" B = l V , 1, a' , 0, c' a", 0, c" , &c. or what is the same thing, the inverse or reciprocal matrix is given by the equation ( a , 6 , c )-^ 1 ( BaV, Ba'V, a„^ V ) a', b\ c' SfrV, a,,.v, a^-v ScV, ac'V, a,..v where of course the differentiations must in every case be performed as if the terms a, 6, &c. were all of them independent arbitrary quantities. 18. The formula shows, what is indeed clear d priori, that the notion of the inverse or reciprocal matrix fails altogether when the determinant vanishes: the matrix is in this case said to be indeterminate, and it must be understood that in the absence of express mention, the particular case in question is frequently excluded from consideration. It may be added that the matrix zero is indeterminate; and that the product of two matrices may be zero, without either of the factors being zero, if only the matrices are one or both of them indeterminate. 19. The notion of the inverse or reciprocal matrix once established, the other negative integer powers of the original matrix are positive integer powers of the inverse or reciprocal matrix, and the theory of such negative integer powers may be taken to be known. The theory of the fractional powers of a matrix will be ftirther discussed in the sequel. 20. The positive integer power L^ of the matrix L may of course be multiplied by any matrix of the same degree: such multiplier, however, is not in general con- vertible with L; and to preserve as far as possible the analogy with ordinary algebraical frmctions, we may restrict the attention to the case where the multiplier is a single quantity, and such convertibility consequently exista We have in this manner a matrix cL^, and by the addition of any number of such terms we obtain a rational and integral function of the matrix L. a 11. 61 482 A MEMOIR ON THE THEORY OF MATRICES. [132 21. The general theorem before referred to will be best anderstood by a com- plete deTelopment of a particular case. Imagine a matrix JI={ a, 6 ), c, d and form the determinant a — M, h c ,d-M the developed expression of this determinant is Jf » - ( a + rf) JT + (ad - 6c) Jf • ; the values of M\ M\ Jf* are ( a^^hc , 6(a + rf) ), ( a. 6 ). ( 1. X I ' i cia-^-d), (f + 6c ' c, d\ ! 0, 1 i and substituting these values the determinant becomes equal to the matrix zero, vis. we have a-Jf, 6 =(a» + ftc, 6(a + ci) )-(a + rf) ( a, 6 ) + (ad-ftc) ( 1, 0) c ,d-M\ I c(a + d), cf + ftc ! c, d ^ 0, 1 I = ( (a» + ftc)-(a + ci)a + (a(i-6c), 6(a + d)-(a + rf)6 ) = ( 0, ); c(a + d)-(a + d)c , cf + ftc-(a + cOd+arf-6c ! ! 0, I that is a - Jf , 6 . = 0, c ,d^M where the matrix of the determinant is ( a, 6 )-i/( 1, ), c, d I I 0, 1 I that is, it is the original matrix, diminished by the same matrix considered as a single quantity involving the matrix unity. And this is the general theorem, viz. the deter- minant, having for its matrix a given matrix less the same matrix considered as a single quantity involving the matrix unity, is equal to zero. 22. The following symbolical representation of the theorem is, I think, worth noticing: let the matrix if, considered as a single quantity, be represented by JB, then writing 1 to denote the matrix unity, S,l will represent the matrix M, considered as a single quantity involving the matrix unity. Upon the like principles of notation, I. if will represent, or may be considered as representing, simply the matrix if, and the theorem is Det (I.if-iBf.l) = 0. 152] A MEMOIR ON THE THEORY OF MATRICES. 483 23. I have verified the theorem, in the next simplest case of a matrix of the order 3, viz. if Jf be such a matrix, suppose if = ( a, 6, c ), d, e, f then the derived determinant vanishes, or we have a — M, b d , e 9 , h > c M, f , i-if = 0, or expanding Jf* — (a + e + i) if ^ + (^' + ta + 06 —/A — eg — bd) M — {ad + bfg + cdh — afh — bdi — ceg) = ; but I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree. 24. If we attend only to the general form of the result, we see that any matrix whatever satisfies an algebraical equation of its own order, which is in many cases the material part of the theorem. 25. It follows at once that every rational and integral function, or indeed every rational function of a matrix, can be expressed as a rational and integral function of an order at most equal to that of the matrix, less unity. But it is important to consider how far or in what sense the like theorem is true with respect to irrational functions of a matrix. If we had only the equation satisfied by the matrix itself, such extension could not be made; but we have besides the equation of the same order satisfied by the irrational function of the matrix, and by means of these two equations, and* the equation by which the irrational function of the matrix is deter- mined, we may express the irrational function as a rational and integral function of the matrix, of an order equal at most to that of the matrix, less unity; such expression will however involve the coefficient of the equation satisfied by the irrational function, which are functions (in number equal to the order of the matrix) of the terms, assumed to be unknown, of the irrational function itself. The transformation is never- theless an important one, as reducing the number of unknown quantities firom n' (if n be the order of the matrix) down to n. To complete the solution, it is necessary to compare the value obtained as above, with the assumed value of the irrational function, which will lead to equations for the determination of the n unknown quantitiea 26. As an illustration, consider the given matrix M=( a, 6 ), c, d 61—2 484 A MEMOIR ON THE THEORY OF MATRICES. [152 and let it be required to find the matrix L = *JM, In this case M satisfies the equation and in like manner if then L satisfies the equation if»-(a + d)Jf + ad-6c = 0; 7. S Z« - (a + S) Z + aS- /87 = ; and from these two equations, and the rationalized equation Z' = M, it should be possible to express L in the form of a linear function of M: in fact, putting in the last equation for Z" its value (=M), we find at once Z = i-g[.lf + (aS-^7)], which is the required expression, involving as it should do the coefficients a + S, aS^/3y of the equation in L. There is no difficulty in completing the solution; write for shortness a + S = X, aS — ySy = F, then we have Z = ( 0, fi ) = ( a'\-Y X b_ ), X d+T and consequently forming the values of a + S and aS — ^87, X = a+d+2Y y_ (a+r)(d+r)-6c and putting also a + d = P, ad — bc^Q, we find without difficulty and the values of a, /8, 7, S are consequently known. The sign of ^Q is the same in both formulae, and there are consequently in all four solutions, that is, the radical vif has four values. 27. To illustrate this further, suppose that instead of M we have the matrix if> = ( a, 6 )» = ( d' + bc , 6(a + d) ), c, d c(a + d), d» + 6c 152] A MEMOIR ON THE THEORY OF MATRICES. 485 so that D = M^, we find P = (aH-d)«-2(ad-6c), and thence VQ = ± (ad — be). Taking the positive sign, we have and these values give simply But taking the negative sign, F= ad — be, X=±(a + d), L=±^( a, b )=±M, I c, d Y=-ad + bCy X = ± V(a - d)« + 46c, and retaining X to denote this radical, we find Z = _( a'-ad+26c b(a + d) ), X c(a'{-d) X d^-ad + 2bc which may also be written , a+d(a, b) 2(ad-6c) ( 1, ), ^ I 0, 1 I or, what is the same thing. c, d J ^a + d ^ 2 (ad — be) " Z X ' and it is easy to verify d posteriori that this value in fact gives D = M\ It may be remarked that if if» = ( 1, )» = 1. 0, 1 the last-mentioned formula fails, for we have X = ; it will be seen presently that the equation Z* = 1 admits of other solutions besides Z = ± 1. The example shows how the values of the fractional powers of a matrix are to be investigated. 28. There is an apparent difficulty connected with the equation satisfied by a matrix, which it is proper to explain. Suppose, as before, M^( a, 6 ), c, d 486 A MEMOIB ON THE THEORY OF MATRICES. [152 so that M satisfies the equation a - If, 6 =0, and let X^, X„ be the single quantities, roots of the equation or c d — X = or Z«-(a+(i)Z + ad-6c=0. The equation satisfied by the matrix may be written (if-Z,)(if-Z„) = 0. in which X,, X,, are to be considered as respectively involving the matrix unity, and it would at first sight seem that we ought to have one of the simple fiEu^tors equal to zero; this is obviously not the case, for such equation would signify that the perfectly indeterminate matrix M was equal to a single quantity, considered as involving the matrix unity. The explanation is that each of the simple factors is an indeterminate matrix, in fact M — X, stands for the matrix (a-X„ b d-X ). and the determinant of this matrix is equal to zero. The product of the two factors is thus equal to zero without either of the factors being equal to zero. 29. A matrix satisfies, we have seen, an equation of its own order, involving the coefficients of the matrix; assume that the matrix is to be determined to satisfy some other equation, the coefficients of which are given single quantities. It would at first sight appear that we might eliminate the matrix between the two equations, and thus obtain an equation which would be the only condition to be satisfied by the terms of the matrix ; this is obviously wrong, for more conditions must be requisite, and we see that if we were then to proceed to complete the solution by finding the value of the matrix common to the two equations, we should find the matrix equal in every case to a single quantity considered as involving the matrix unity, which it is clear ought not to be the case. The explanation is similar to that of the difficulty before adverted to; the equations may contain one, and only one, common factor, and may be both of them satisfied, and yet the common factor may not vanish. The necessary condition seems to be, that the one equation should be a factor of the other; in the case where the assumed equation is of an order equal or superior to the matrix, then if this equation contain as a factor the equation which is always satisfied by the matrix, the assumed equation will be satisfied identically, and the condition is sufficient as well as necessary: in the other case, where the assumed equation is of an order inferior to that of the matrix, the condition is necessary, but it is not sufficient. 152] A MEMOIR ON THE THEORY OF MATRICES. 487 30. The equation satisfied by the matrix may be of the form M^ = 1 ; the matrix is in this ease said to be periodic of the ?ith order. The preceding conside- rations apply to the theory of periodic matrices; thus, for instance, suppose it is required to find a matrix of the order 2, which is periodic of the second order. Writing Jf=( a, 6 ), c, d we have and the assumed equation is Jlf»-(a + d)if+a(i-6c = 0, :ilf « - 1 = 0. These equations will be identical if a + d = 0, a(i - 6c = — 1, that is, these conditions being satisfied, the equation Jf * — 1 = required to be satisfied, will be identical with the equation which is always satisfied, and will therefore itself be satisfied. And in like manner the matrix M of the order 2 will satisfy the condition Jf'— 1 = 0, or will be periodic of the third order, if only Jf'— 1 contains as a factor if " - (a + d) -M'+ a(i - 6c, and so on. 31. But suppose it is required to find a matrix of the order 3, M^{ a, d, 6, c) e. f which shall be periodic of the second order. Writing for shortness a — Jf, h , M, f ^-iM^-AM^ + BM^C), the matrix here satisfies M*-AM* + BM-C = 0, and, as before, the assumed equation is Jf ' — 1 = 0. Here, if we have 1+J5 = 0, A + C=^0, the left-hand side will contain the £Ekctor (if'— 1), and the equation will take the form (ilf* — 1) (if + C) = 0, and we should have then if' —1 = 0, provided M+C were not an indeterminate matrix. But M+C denotes the matrix ( a+C, b ) d 9 e + C. f h , i + C 488 A MEMOIR ON THE THEORY OF MATRICES. [152 the determinant of which is C* + -4C*+J5(7+ C, which is equal to zero in virtue of the equations 1+-B = 0, -44-C = 0, and we cannot, therefore, from the equation (if»-l)(if+C) = 0, deduce the equation if«-l = 0. This is as it should be, for the two conditions are not sufficient, in fact the equation Jf« = ( a^ +bd +cgy ab + be + ch, ac + bf+ d ) da + ed +fg, db + ei^ -^fh, dc + ef-¥fi ga +hd + ig, gb + he + ih, gc + hf+ i* = 1 gives nine equations, which are however satisfied by the following values, involving in reality four arbitrary coefficients; viz. the value of the matrix is ( - (7 -f g) fiv-' a + /8 + 7 a + /8 + 7 ^(0 + y)vfl'" ^(0+y)vfjr^ ) a + /8 + 7 - (7 + g) Xfi-^ a + fi + y o + i8 + 7 /9 a + 13 + y - (0 + /3) I/X-' 04-/8 + 7 a + /8 + 7 so that there are in all five relations (and not only two) between the coefficients of the matrix. 32. Instead of the equation M^ — 1=0, which belongs to a periodic matrix, it is in many cases more convenient, and it is much the same thing to consider an equation M^ — k = 0, where A; is a single quantity; The matrix may in this case be said to be periodic to a factor pris. 33. Two matrices L, M are convertible when LM = ML. If the matrix M is given, this equality affords a set of linear equations between the coefficients of L equal in number to these coefficients, but these equations cannot be all independent, for it is clear that if X be any rational and integral function of M (the coefficients being single quantities), then L will be convertible with Jf ; or what is apparently (but only appa- rently) more general, if L be any algebraical function whatever of M (the coefficients being always single quantities), then L will be convertible with M. But whatever the form of the function is, it may be reduced to a rational and integral function of an order equal to that of M, less unity, and we have thus the general expression for the matrices convertible with a given matrix, viz. any such matrix is a rational and integral function (the coefficients being single quantities) of the given matrix, the order being that of the given matrix, less unity. In particular, the general form of the matrix L convertible with a given matrix M of the order 2, is Z = aAf + /8, or what is the same thing, the matrices ( a, 6 ), ( a\ V ) c, d c', d' will be convertible if a' — d' : 6' : c' = a — d : 6 : c. 152] A MEMOIR ON THE THEORY OF MATRICES, 489 34. Two matrices Z, M are skew convertible when LM = — ML ; this is a relation much less important than ordinary convertibility, for it is to be noticed that we cannot in general find a matrix L skew convertible with a given matrix M. In fact, con- sidering M as given, the equality affords a set of linear equations between the coeffi- cients of L equal in number to these coefficients; and in this case the equations are independent, and we may eliminate all the coefficients of Z, and we thus arrive at a relation which must be satisfied by the coefficients of the given matrix M, Thus, suppose the matrices ( a, 5 ), ( a\ V ) c, c\ d! are skew convertible, we have (a, h){a\ 6' ) = ( oa' + fcc', oJ' + W ), 0, d \ d y dl ca' + dc', cV + dd! (a\ 5' )( a, 5) = (oa' + 6'c, a'5+6'rf), c\ d! \ c, d and the conditions of skew convertibility are c'a + d'c, c'b + d'd 2oa' + 5c' + b'c = 0, 5' (a +rf)+6(a'+d') = 0, c'{a +d) +c(a' + dO = 0» 2dd' + bc' + b'c =0. Eliminating a\ b\ c\ d\ the relation between a, 5, c, d is 2a. c , b , . b , a +d, b c , • a + d, c • c , b , 2d = 0, which is (a + dy (ad-bc)-^ 0. Excluding from consideration the case orf — 6c = 0, which would imply that the matrix was indeterminate, we have a + d = 0. The resulting system of conditions then is a + d = 0, a' + d' = 0, aa' + 6c' + 5'c + dd' = 0, the first two of which imply that the matrices are respectively periodic of the second order to a factor pris, 35. It may be noticed that if the compound matrices LM and ML are similar, they are either equal or else opposite ; that is, the matrices L, M are either convertible or skew, convertible. C. II. 62 490 A MEMOIR ON THE THEORY OF MATRICES. [152 36. Two matrices such as ( a, 6 ), ( a, c ), c, d 6, d are said to be formed one fix)m the other by transposition, and this may be denoted by the symbol tr. ; thus we may write (a, c ) = tr. ( a, 6 ). 6, d \ c, d The effect of two successive transpositions is of course to reproduce the original matrix. 37. It is easy to see that if M be any matrix, then (tr. My = tr. (MP), (tr. M)-' = tr. ( Jf-0. and in particular, 38. If Z, if be any two matrices, tr. (Zi/) = tr. il/. tr.Z, and similarly for three or more matrices, L, M, N, &c., tr. (LMN) == tr. N. tr. M, tr. i, &c. 40. A matrix such as (a, A, g ) ' K 6, / which is not altered by transposition, is said to be symmetrical. 41. A matrix such as ( 0, V, -m) -V, 0, X I A^ -X, ! which by transposition is changed into its opposite, is said to be skew symmetrical. 42. It is easy to see that any matrix whatever may be expressed as the sum of a s}rmmetrical matrix, and a skew sjrmmetrical matrix ; thus the* form h-v, 6., / + X flr + M, /-X, c which may obviously represent any matrix whatever of the order 3, is the sum of the two matrices last before mentioned. 152] A MEMOIB ON THE THEORY OF MATRICES. 491 43. The following formulae, although little more than examples of the composition transposed matrices, may be noticed, viz. (a, 6 $ a, c ) = ( a' + 6', ac-{-bd ) c, d d, 5 ac-^bd, c' + d' which shows that a matrix compounded with the transposed matrix gives rise to a symmetrical matrix. It does not however follow, nor is it the fact, that the matrix and transposed matrix are convertible. And also (a, c ^ a, 5 $ a, c ) = ( a» + 6cd + a(6» + c'), c» +a6d + c(a« + cP) ) b, d c, d \ \ b, d which is a remarkably symmetrical form. It is needless to proceed further, since it is clear that (a, c ^ a, 6 $ «> ^ a, 6 )=(( a, c 1^ a, b ))*. b, d \ c, d b, d c, d b, d Cf d 44. In all that precedes, the matrix of the order 2 has frequently been con- sidered, but chiefly by way of illustration of the general theory ; but it is worth while to develope more particularly the theory of such matrix. I call to mind the fundamental properties which have been obtained, viz. it was shown that the matrix satisfies the equation and that the two matrices M = ( a. 5 ), 1 c, a ! M*-{a + d)M + ad-bc = 0, (a, b). ( a\ V ), c, d will be convertible if c', d' a! — d' : V ', c' ^a^d : b : Ct and that they will be skew convertible if the first two of these equations being the conditions in order that the two matrices may be respectively periodic of the second order to a factor jyrhs, 45. It may be noticed in passing, that if Z, Jlf are skew convertible matrices of the order 2, and if these matrices are also such that i> = — 1, ilf' = — 1, then putting JV=Z3f= — JfL, we obtain Z«=-l. if«=-l, i<r« = -i, L^MN^^NM, M^^NL^^NL, N^LM^-ML, which is a system of relations precisely similar to that in the theory of quatemiona 62—2 492 A MEMOIR ON THE THEORY OF MATRICES. [152 46. The integer powers of the matrix Jf = ( a, 5 \ c, d are obtained with great facility from the quadratic equation ; thus we have, attending first to the positive powers, Jtfa = (a 4- d)M-(ad - be), iV» = [(a + d)« - (orf - 6c)] if - (a + d) (orf - 6c), &c., whence also the conditions in order that the matrix may be to a factor pris periodic of the orders 2, 3, &c. are a + d * = 0, (a + dy - (ad - 6c) = 0, &c. ; and for the negative powers we have (od- 6c) -¥-' = - il/ + (a + d), which is equivalent to the ordinary form {ad''bc)M-' = { . d, -6 ); -c, a I and the other negative powers of M can then be obtained by successive multiplications with M-\ 47. The expression for the nth power is however most readily obtained by means of a particular algorithm for matrices of the order 2. Let A, 6, c, J, q he any quantities, and write for shortness iJ = — A' — 46c; suppose also that h\ h\ c', tT, q' are any other quantities, such nevertheless that A' : 6' : c' = A : 6 : c, and write in like manner iJ* = — A'* — 46V. Then observing that —r^ . -—r . — are respectively equal to -f=^,> "7=57 » "7^* *^^ matrix { j( , h\ 2hJ ) 2cJ 'JR .J (cot, + -4) contains only the quantities J, q, which are not the same in both systems; and we may therefore represent this matrix by (J, q\ and the corresponding matrix with 152] A MEMOIR ON THE THEORY OF MATRICES. 493 h\ h\ c\ J\ q' by {J\ q'). The two matrices are at once seen to be convertible (the assumed relations h' : h' : c' — h : h : c correspond in fact to the conditions, a'^d! : b* : c' = a — d : b : c, of convertibility for the ordinary form), and the compound matrix is found to be \sin or sm a ^ ^ / and in like manner the several convertible matrices {J, q), (J\ q'\ (J'\ q") &c. give the compound matrix Vsinasma smg ... ^ ^ i j 48. The convertible matrices may be given in the first instance in the ordinary^ form, or we may take these matrices to be ( a, 6 ), ( a\ V ), ( a", 6" ), &c. c, d c\ d' c\ d" where of course d — a : b : c = d* — a' : b' : cf = d" — a" : b'* : c** ^ &c. Here writing h = d — a, and consequently R = — {d — o,y — 46c, and assuming also «/ = J VS and cot a = — , — , and in like manner for the accented letters, the several matrices are respectively (i -^R. q) (4 >^, q'), (i ViP'. q"), &c.. and the compound matrix is / 8in( «y + q' + q" ^ ^^ ^g> ^ ^ + ,' + y"+ ...) . 49. When the several matrices are each of them equal to (a, 5 ), c, d we have of course g = g' = g" . . . , R = R = Itf\,, , and we find ( a, 6 )- c, c2 =(s«^. "5)^ or substituting for the right-hand side, the matrix represented by this notation, and putting for greater simplicity ^p (i v:r)» = (J m L,oT L= ^l^i"? (i vs)»-. 494 A MEMOIR ON THE THEORY OF BfATRICE& [152 we find ( a. b )»=:(iZ(v^cotn9-(d-a)), U ) \ c, d \ Lc , i Z (Vii cot wj + (d — a)) ; where it will be remembered that -B = — (d — a)* — 46c and cot 9 = ._ , the last of which equations may be replaced by . J — 1 . d + a + V^TS cosg + v — l8ma= 7- , ^ ^ 2Vad-6c The formula in £Ekct extends to negative or fractional values of the index n, and when n is a fraction, we must, as usual, in order to exhibit the formula in its proper generality, write q + 2nnr instead of q. In the particular case n » ^, it would be easjr to show the identity of the value of the square root of the matrix with that before obtained by a different process. 50. The matrix will be to a factor pris, periodic of the nth order if only sinitf^O, that is, if 9 = - (m must be prime to n, for if it were not, the order of periodicity would be not n itself, but a submultiple of n) ; but cos q = — . , and the condition 2 V od — be is therefore (d + o)»- 4 (ad -6c) COS' "^ = 0, or as this may also be written, (P-^a^^ 2ad cos h 46c cos* — = 0, n n a result which agrees with those before obtained for the particular values 2 and 3 of the index of periodicity. 51. I may remark that the last preceding investigations are intimately connected QX -\~ h with the investigations of Babbage and others in relation to the function <f)X = -^ . I conclude with some remarks upon rectangular matrices. 52. A matrix such as ( a, 6, c ) I a', 6', c I where the number of columns exceeds the number of lines, is said to be a broad matrix ; a matrix such as ( a, 6 ) where the number of lines exceeds the number of columns, is said to be a deep matrix. 152j A MEMOIR ON THE THEORY OF MATRICES. 495 53. The matrix zero subsists in the present theory, but not the matrix unity. Matrices may be added or subtracted when the number of the lines and the number of the columns of the one matrix are respectively equal to the number of the lines and the number of the columns of the other matrix, and under the like condition any number of matrices may be added together. Two matrices may be equal or opposite, the one to the other. A matrix may be multiplied by a single quantity, giving rise to a matrix of the same form ; two matrices so related are similar to each other. 54. The notion of composition applies to rectangular matrices, but it is necessary that the number of lines in the second or nearer component matrix should be equal to the number of columns in the first or further component matrix; the compound matrix will then have as many lines as the first or further component matrix, and as many columns as the second or nearer component matrix. 55. As examples of the composition of rectangular matrices, we have {a.b,cji a\ b\ c\ d' ) = ( (a, 5, cJia\ e\ iO> («. b, c$h\f\ f) (a, 6, cl^c^ g\ k'\ (a, 6, c$d', K V) ), d,ej e\f\g\K (d, e,/$< e\ i% (d, e,/$6', /, /) (d, e,/$c', fir', ^ (d, 6,/K. K V) and ( a, d $ a', V, d, d') ) = ( (a. d$a'. e'). («. ^F. A («. ^<^> 9'\ («. ^d', h') ). (6, e$a'. e'), (h, eji'./), (6. e$c'. (/'), {h. ejd', h!) (c, /$a'. e'), (c. /$6', /), (c, /$c'. 9')> (c /K. A') c./l 56. In the particular case where the lines and columns of the one component matrix are respectively equal in number to the columns and lines of the other com- ponent matrix, the compound matrix is square, thus we have ( o, 6, c $ a', d' ) = ( (a. h, c$o', h', c'), (a, h, c^d', e', f) ) d, e, / V, e' (d. e. f^a', b', cO, (d, e, f^d', e', /') and ( a', d' $ a, 6, c ) = ( (a', d'$a, d), (a', d'$6, e), (a', d'^c, f) ). V, e' c'.f d, e,f {V, ii \a, d). (6', e' $6, e), (6', c' $c, /) (c', f\a, d), (c', f'\h, e). (c'. /'$c, /) The two matrices in the case last considered admit of composition in the two different orders of arrangement, but as the resulting square matrices are not of the same order, the notion of the convertibility of two matrices does not apply even to the case in question. 57. Since a rectangular matrix cannot be compounded with itself, the notions of the inverse or reciprocal matrix and of the powers of the matrix and the whole resulting theory of the functions of a matrix, do not apply to rectangular matrices. 496 A MEMOIR ON THE THEORY OF MATRICES. [152 58. The notion of transposition and the symbol tr. apply to rectangular matrices, the effect of a transposition being to convert a broad matrix into a deep one and reciprocally. It may be noticed that the symbol tr. may be used for the purpose of expressing the law of composition of square or rectangular matrices. Thus treating (a, 6, c) as a rectangular matrix, or representing it by (a, 6, c), we have tr. ( a\ b\ c' ) = ( a ), and thence ( a, b, c ) tr. ( a\ h\ c' ) = (a, 6, cY a' ) = (a, 6, c\a\ h\ c'), so that the symbol (a, 6, c\a\ b\ c*) would upon principle be replaced by ( a, 5, c ) tr. (a', 5', c' ): I III it is however more convenient to retain the symbol (a, 6, c^a\ b\ c^). Hence introducing' the symbol tr. only on the left-hand sides, we have ^ a, 6, c ) tr. ( a, b\ cM = ( (a, 6, cja\ b\ c'), (a, 6. c$d', e\ f) ), d, ^. / I i d\ ^, /' I I (d. e, fla\ b\ c% (d, e, fJid\ ^. /O | or to take an example involving square matrices. ( a, 5 ) tr. ( a', 6' ) = ( (a, 6$a', 5'), (a, 6$d', e') ) ; I d, e I I d', e' I I (d, c$a', 6'), (d, e^d\ e') it thus appears that in the composition of matrices (square or rectangular), when the second or nearer component matrix is expressed as a matrix preceded by the symbol tr., any line of the compound matrix is obtained by compounding the corresponding line of the first or further component matrix successively with the several lines of the matrix which preceded by tr. gives the second or nearer component matrix. It is clear that the terms 'symmetrical* and *skew symmetrical* do not apply to rectangular matrices. 153] 497 153. A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION OF A BIPARTITE QUADRIC FUNCTION. [From the Philosophical Transactions of the Royal Society of London, vol. cxLvm. for the year 1858, pp. 39 — 4!6. Received December 10, 1857, — Read January 14, 1858.] The question of the automorphic linear transformation of the function a^ •\- y^ •¥ s^, that is the transformation by linear substitutions, of this function into a function a?/ + y/ + V of the same form, is in eflfect solved by some formulae of Euler's for the transformation of coordinates, and it was by these formulae that I was led to the solution in the case of the sum of n squares, given in my paper "Sur quelques pro- pri^tds des determinants gauches"(0- ^ solution grounded upon an d priori investiga- tion and for the case of any quadric function of n variables, was first obtained by M. Hermite in the memoir "Remarques sur une M^moire de M. Cayley relatif aux determinants gauches"(')- This solution is in my Memoir "Sur la transformation d'une function quadratique en elle-meme par des substitutions lin6aires"('), presented under a somewhat different form involving the notation of matrices. I have since found that there is a like transformation of a bipartite quadric function, that is a lineo-linear function of two distinct sets, each of the same number of variables, and the develop- ment of the transformation is the subject of the present memoir. 1. For convenience, the number of variables is in the analytical formulae taken to be 3, but it will be at once obvious that the formulae apply to any number of variables whatever. Consider the bipartite quadric ( a , 5 , c $a?, y, ^][x, y, z), a' , 6' , c' a y , c 1 CreUe, X. xxxn. (1846) pp. 119—123, [62]. ' Cambridge and Dublin Mathematical Journal^ t. ix. (1854) pp. 63 — 67. » CrelU, t. L. (1866) pp. 288—299, [186]. c. II. 63 498 A MEMOm ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153 which stands for (ax + 6y -hcz )x + (a'x + Vy +c'z)y + (a"a? + V'y + c''z) z, and in which {x^ y, z) are said to be the nearer variables, and (z, y, z) the further variables of the bipartite. 2. It is clear that we have ( a , 6 , c $a?, y, ^$x, y, z) = ( a, a\ a" $x, y, z$a?, y, z) «'' j»'' -•" a , , c 6, y, 6" c, c , c // and the new form on the right-band side of the equation may also be written (tr. ( a , 6 , c ) $x, y, z^x, y, z), ^" !»" V a , , c that is, the two sets of variables may be interchanged, provided that the matrix is transposed 3. Each set of variables may be linearly transformed: suppose that the substitu- tions are («, y, -8^) = ( i , wi , n $a?^, y,, z,) and (X, y. z) = ( 1 . 1' . 1" $^,. y,. O- // m, m, m n , n , n Then first substituting for {x, y, z) their values in terms of (a?^, y^, z^\ the bipartite becomes ( ( a , 6 , c 5^ Z , m , n ) $a?„ y„ ^,$x, y, z); a", 5", c" represent for a moment this expression by ( A , B , C $a?,, y,, z,^x, y, z), 153] OF A BIPARTITE QUADRIC FUNCTION. 499 then substituting for (x, y, z) their values in terms of (x^, y^, z^), it is easy to see that the expression becomes ( ( 1 , m , n $ il , 5 , ) $/c„ y,, zjx„ y„ z,), r, m'', n' A\ R, cr and re-establishing the value of the auxiliary matrix, we obtain, as the final result of the substitutions, ( a , 5 , c $a?, y, «$x, y, z) = (( 1 , m , n ISa ,h , c \l , m , n ) $a?,, y,, ^^$x,, y,, zj, _// L/' >." r, m', n' r, m", n' a', 6', c' _// L// -// a , , c that is, the matrix of the transformed bipartite is obtained by compounding in order, first or furthest the transposed matrix of substitution of the further variables, next the matrix of the bipartite, and last or nearest the matrix of substitution of the nearer variablea 4. Suppose now that it is required to find the automorphic linear transformation of the bipartite ( a , 6 , c\ X, y, z^, y, z), a' , V y c' ^" u* V a , , c or as it will henceforward for shortness be written, (n$a?, y, ^$x, y, z); this may be eflfected by a method precisely similar to that employed by M. Hermite for an ordinary quadric. For this purpose write a? + a?^ = 2f, y + y, = 2i7, z + ^, = 2f, x + x, = 2B, y + y, = 2H, z + z^ = 2Z, or, as these equations may be represented, (a?+a:„ y-)ry,, z + -f,)= 2(f , 17 , f ), (x + x„ y + y,, z + z^)=2(H, H, Z); then we ought to have (n$2f-a?, 217 -y, 2f-.^][2a-x, 2H-y, 2Z - z) = (n][a;, y, -erjx, y, z). 5. The left-hand side is 4(ft$f, ^, ?$H, H, Z)-2(n$a?, y, ^$B, H, Z)-2(n$f 17, ?$x, y, z) + n$^, y, ^$x, y, z), and the equation becomes 2(n$f 1,. r$B. H. Z)-(fl$a:, y, r$S. H. Z)-(ft$f. ,,. r$x, y. z)=:0, 63—2 500 A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153 or as it may be written, ("$f , V. ?$B, H, Z) -. (a^x, y, ^$B, H, Z) . y. z)) -X, H-y. Z-z){ ' -X. v-y. ?-«$a. H, Z)]^ -X, H-y. Z-z$f. ,. ?)) ' + (n$f. V. ?$B. H, z) - (n$f, ,, r$x or acfain, ^' (n$f-.. ,-y, ?-.$B. H. Z) + («$?. I?. ?$B or what is the same thing, (ft$f-^, v-y. ?-^$H, H, Z) + (tr. njB and it is easy to see that the equation will be satisfied by writing (tr.n$H-x, H-y, Z-z) = -(tr.T$B, H, Z). where T is any arbitrary matrix. In fact we have then ( n$f -X, V -y, K -^$B. H, Z)= ( T$f , , , r$B, H. Z), (tr.n$S-x. H-y, Z-z$f . i, , r) = -(tr.T$H, H, Z$f , ,,. f) ( T$f , ^ , ?$B. H, Z), and the sum of the two terms consequently vanishe& 6. The equation gives (n-T$f, ,, K)=mj'.y.'). and we then have In fact the two equations give or what is the same thing, which is the equation assumed as the definition of (^, 17, ^); and conversely, this equation, combined with either of the two equations, gives the other of them. 7. We have consequently (^, y, z) = (n-^ (ft -. T)$f , 17. ?), (f, 17, ?) = ((" + T)-^"$^., y.»0» and thence (X. y, *)=(n->(n-T)(n+T)->n$^„ y„ o. 153] OF A BIPARTITE QUADRIC FUNCTION. 501 8. But in like maimer the equation (tr.n$H-x, H-y, Z - z) = - (tr. T$B, H, Z) gives (tr. fl + T$E, H, Z) = (tr. n$x, y, z), and we then obtain (tr.fl^TJB, H, Z) = (tr.ft$x„ y,, z,). 9. In tact these equations give (tr.2n$B, H, Z) = (tr.fl$x + x„ y + y„ z + z^), or 2(a. H, Z) = (x + x,, y + y„ z + z,); and conversely, this equation, combined with either of the two equations, gives the other of them. We have then (x, y , z) = ((tr.ft)-ntr.fl + T$H, H, Z), (B, H, Z) = ((tr.n-T)-^ tr. n$x,, y,, z> and thence (X, y, z) = ((tr. ft)-» (tr. flTT)(tr. fl-T)-Hr. ftjx,, y„ z,). 10. Hence, recapitulating, we have the following theorem for the automorphic linear transformation of the bipartite (n$a?, y, z^x, y, z), viz. T being an arbitrary matrix, if (x, y, ^)=:(fl-nn-T)(fl + T)-^n$a;,, y„ O, (x, y, z) = ((tr.n)-Mtr. n + T)(tr. fl-T)-»tr. n$x„ y„ z,), then (ft$a?, y, ^$x, y, z) = (n$a?,, y„ s,$x„ y„ z), which is the theorem in question. 11. I have thought it worth while to preserve the foregoing investigation, but the most simple demonstration is the verification d posteriori by the actual substitution of the transformed values of (x, y, z), (x, y, z). To effect this, recollecting that in general tr. (il~*) = (tr. -4)"* and tr. ABCD ^ tr, D. tr. C, tr. B. tr. A, the transposed matrix of substitution for the further variables is ft (ft - T)-Hn + T) n-^ and compounding this with the matrix ft of the bipartite, and the matrix ft-Mft-T)(ft+T)-^ft 502 A MEMOIR ON THE AUTOMOBPHIC LINEAR TRANSFORMATION [l53 of substitution for the nearer variables, the theorem will be verified if the result k equal to the matrix fl of the bipartite ; that is, we ought to have or what is the same thing, 11(1:1 - T)-Hft + T) n-Hfl - T)(ft + T)-^fl = n ; this is successively reducible to (n -h T)ft->(n - T) = (ft - T)ft-'(ft + T), ft-^ft + T)ft-»(ft - T) = ft-^ft - T)ft-'(ft + T), (1 + ft-'T)(l - ft-^T) =(l-.ft-*T)(l + fl-^T), which is a mere identity, and the theorem is thus shown to be true. 12. It is to be observed that, in the general theorem, the transformations or matrices of substitution for the two sets of variables respectively are not identical, but it may be required that this shall be so. Consider first the case where the matrix ft is sjnumetrical, the necessary condition is that the matrix T shall be skew symmetrical ; in fact we have then tr. ft = ft, tr. T=-T, and the transformations become (x, y, ^) = (ft-Hft-T)(ft+T)-»ft$a:,, y„ z,\ (X, y, z) = (ft-^(ft - T)(ft + T)-^ft$x„ y„ z,), which are identical. We may in this case suppose that the two sets of variables become equal, and we have then the theorem for the automorphic linear transformation of the ordinary quadric (ft$a?, y, zy, viz. T being a skew sjnumetrical matrix, if (x, y, ^) = (ft-^n - T)(ft + T)-^ft$a?„ y„ z,), then (ft$a?, y, zy = (ft$a;„ y„ z,y, 13. Next, if the matrix ft be skew symmetrical, the condition is that the matrix T shall be sjnumetrical ; we have in this case tr. ft = — ft, tr. T = T, and the four factors in the matrix of substitution for (x, y, z) are respectively — ft~*, — (ft — T), — (ft + T)-* and —ft, and such matrix of substitution becomes therefore, as before, identical with that for (a?, y, z)\ we have therefore the following theorem for the automorphic linear transformation of a skew symmetrical bipartite (ft$a?, y, «$x, y, z), 153] OF A BIPARTITE QUADRIC FUNCTION. 503 when the transformations for the two sets of variables are identical, viz. T being any symmetrical matrix, if {X. y. ^) = (fi-'(n-T)(n + T)-'n$«,. y„ z). (X, y, z) = (n->(fi-T)(n + T)->ft$x„ y„ z,). then (ft$a?, y, £$x, y, z)=(ft$aj,, y„ z,$x,, y„ z,). 14. Lastly, in the general case where the matrix H is anything whatever, the condition is fl-^T = - (tr. n)-* tr. T for assuming this equation, then first n-^n - T) = (tr. n)-Htr. fn- t), and in like manner Qr\a + T) = (tr. n)-*(tr. n - T). But we have 1 = (tr. n)-i(tr. n - T) (tr. n - T)-» tr. n, and therefore, secondly, (ft + T)-* ft = (tr. ft - T)-* tr. ft ; and thence ft-^n - T)(ft + T)-»ft = (tr. ft)-» (tr. ft + T) (tr. ft - T)-Hr. ft, or the two transformations are identical. 15. To further develope this result, let ft"* be expressed as the sum of a symmetrical matrix Q^ and a skew symmetrical matrix Q^, and let T be expressed in like manner as the sum of a symmetrical matrix T^ and a skew symmetrical matrix T^. We have then (tr.ft)-* = tr.(ft-0 = eo-e., T =T, + T„ tr. T =To-T„ and the condition, ft"* T = — (tr. ft)-* tr. T, becomes that is, <2.T, + Q,T, = 0, and we have 504 A MEMOIR ON THE AUTOMORPHIC LIXEAB TRANSFORBCAHON [153 or as we may, write it. and thence To = - (i{n-» + tr. ft-»})-»(J{ft-^ - tr. n-»})T,. T = - (i{fl-^ + tr. ft-»})-Ki{«-* - tr. fl-»})T, + T,. where T^ is an arbitrary skew sjnmmetrical matrix. 16. This includes the before-mentioned special cases; first, if fi is symmetrical, then we have simply T^T^, an arbitrary skew symmetrical matrix, which is right Next, if fl is skew symmetrical, then T = — 0-*n~*T^ + T^, which can only be finite for T^ = 0, that is, we have T = -O-*n"*0, and (the first part of T being always sjnumetrical) this represents an arbitrary s}'mmetrical matrix. The mode in which this happens will be best seen by an exampla Suppose fl-i = ( A , jy+i'), tr. ft-»=( A , H^v\ and write then we have T, = ( 0, 6), -0,0 r=-{A, ff)->( 0. !.)( 0, e)+( 0. 0) H, b\ -i;, o! I-^. 0' 1-^, v0 ±jn{-B, H) + ( 0, 0) AB-H H, -A \-0, _ ( vB0 AB-H* -vH0 ^^) + ( -pH0 AB-H ,-^. AB-H' vA0 AB-H* 0, 0) 0, When n is skew symmetrical. A, B, H vanish; but since their ratios remain arbitrary, we may write kA, kB, kH for A, B, H, and assume ultimately « = 0. Writing k0 in the place of 0, and then putting k= 0, the matrix becomes ( vB0 -vB0 ) AB-H*' AB-H* -vH0 vA0 AB-H*' AB-H* which, inasmuch aa A -.0, B : 0, and C : remain arbitrary, represents, as it should do, an arbitrary symmetrical matrix. 153] OF A BIPARTITE QUADRIC FUNCTION. 505 17. Hence, finally, we have the foUoMring Theorem for the automorphic linear transformation of the bipartite qnadric, (ft$a?, y, z^x, y, z), when the two transformations are identical, viz. if T^ be a skew symmetrical matrix, and if T = - (i{n-^ + tr. fl-^})(J{n-^ - tr. fl-^})T, + T, ; then if (X. y, «) = (fl->(n-T)(n + T)->n$ar„ y,. z), (X, y, z) = (fl-> (n - T) (ft + T)-> n$x„ y,. z,); we have (n$a?, y, ^$x, y, z) = (n$a?„ y„ z,^x^, y„ z,); and in particular. If n is a sjnnmetrical matrix, then T is an arbitrary skew sjrmmetrical matrix ; If n is a skew symmetrical matrix, then T is an arbitrary symmetrical matrix. c. n. 64 506 [154 154. SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. [From the PhilosophiccU Transactions of the Royal Society of London^ vol. cxlviil for the year 1868, pp. 47—52. Received March 19,— Read June 18, 1857.] The general formula given at the conclusion of my memoir, "Researches on the Partition of Numbers "(0> ^ somewhat diflferent ftom the corresponding formula of Professor Sylvester', and leads more directly to the actual expression for the number of partitions, in the form made use of in my memoir; to complete my former researches, I propose to explain the mode of obtaining from the formula the expression for the number of partitions. The formula referred to is as follows, viz. if ^ be a rational fraction, the denomi- fx nator of which is made up of &ctors (the same or different) of the form 1—^, and if a is a divisor of one or more of the indices m, and k is the number of indices of which it is a divisor, then where 1 Philotophical Transactions, torn, oxlvi. (1S56) p. 127, [140]. ' Professor Sylvester's researches are published in the Quarterly Mathematical Journal, torn. i. [1857, pp. 141 — 152]; there are some numerioal errors in his value of P (1, 2, 8, 4, 5, 6) q. 154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OP NUMBERS. 507 in which formula [1 — of*] denotes the irreducible &ctor of 1 — o^, that is, the fS^^tor which equated to zero gives the prime roots, and /d is a root of the equation [1 — a?*] = ; the summation of course extends to all the roots of the equation. The index 8 extends from « = 1 to 8=^k; and we have then the portion of the fraction depending on the denominator [1— «"]. In the partition of numbers, we have ^ = 1, and the formula becomes therefore { 7-1 ='... + rT7^i^('^'y~'S^^^■^ /«j[,_«.j n(«-i)^ p-x where Uis-'l)^ " [l^a^y We may write vp = coeff. - in i^^ ^. .. . ^^ t Ape-') /a? = n (1 - ««), where m has a given series of values the same or diflferent. The indices not divisible by a may be represented by m, the other indices by ap, we have then where the number of indices ap is equal to k. Hence f(pe^) = n (1 - p^'er^) H (1 - p^er^p^y, or since /> is a root of [! — «:*] = 0, and therefore />* = 1, we have fipe^) = n (1 - p^e-^) n (1 - er^J^)\ and it may be remarked that if n = i/ (mod. a), where v<a^ then instead of p** we may write p", a change which may be made at once, or at the end of the process of development. We have consequently to find >^ = ^^^i^^^'n(i-.pne-on(i-e-i^)- The development of a fS^^tor ^ ^^ is at once deduced from that of . __ . , and is a series of positive powers of t The development of a factor | _^^ap< ^^ deduced from 1 . 1 * that of ;j — -nj, and contains a term involving -. Hence we have n (1 -p'e-"*) n (1 - e^) =""*-* ««'^^-*-"F5-'^'*-?'^'*'''*'^' and thence XP = P^-*- 64—2 508 SUPPLEMENTABT BESEABCHES ON THE PARTITION OF NUMBEBa [154 The actual development, when k is small (for instance k^\ or £ = 2), is most readily obtained by developing each &ctor separately and taking the product. To do this we have where by a general theorem for the expansion of any function of e*, the coefficient of ^-^ is =ty 1 (K n/i-ca+A)"^ n/ vi-c • (i-c)» — • (i-cy+^ <where as usual A(V = l>'-0/ A«0/ = 2/-2 . l-^ + CK, &c.) and 1 111. 1^. l«p l-6-« e"^2 12 720 30240 where, except the constant term, the series contains odd powers only and the coef- l—y^^Bf .111 ficient of t^'-^ is n9/' ^ ^*' ^^' ^^'" ^^^^^^^8 *^® series ^, ^txi tS"- ^^ Bernoulli's numbers. But when k is larger, it is convenient to obtain the development of the fraction firom that of the logarithm, the logarithm of the fraction being equal to the sum of the logarithms of the simple factors, and these being found by means of the formulae The fraction is thus expressed in the form n(i-.p«)n(ap)f* and by developing the exponential we obtain, as before, the series commencing with Besuming now the formula XP = P^-*> which gives p^ as a function of p, we have Ox [1 - a^] p - a? ' 154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 509 but this equation gives and we have [l-aj«] = (a?-p)(a?-p««)...(a?-.p«a), if 1, Oj, ... a. are the integers less than a and prime to it (a is of course the degree of [1 - ic*]). Hence and therefore or putting for yp its value ^p = - p« n (1 - p«'-0 A^, where a is the degree of [I— a^] and o^ denotes in succession the integers (exclusive of unity) less than a and prime to it. The function on the right hand, by means of the equation [I— p*] = 0, may be reduced to an integral function of p of the degree a — 1, and then by simply changing p into x we have the required function 0x, The 0x fraction ^i — I5T ^^° then by multiplication of the terms by the proper factor be reduced to a fraction with the denominator 1 — a;*, and the coefficients of the numerator of this fraction are the coefficients of the corresponding prime circulator ( ) per a^. Thus, let it be required to find the terms depending on the denominator [1 — a:*] in (l-a?)(l-.a:«)(l-a;»)(l-a?*)(l-a?»)(l-a;«)' these are where p—x p—x and fipe-') (1 - per") (1 - p»6-^) (1 - p'ir^) (1 - p»r^) (1 - O (1 - e-^) 510 SUFPLEMEMTABT RESEARCHES ON THE PARTITIOK OF NUMBEBS. [15^ where it is easy to see that 1 1 A^ = 18il-p)(l-p')(l-p^)(l-p»)' A 1 fl 1 f p ip' 4,p* ^p^ \] ^-'~(l-p)(l-p')(l-f^){l-p')\* 18\l-p^l-f^'^l-p*^l-p^)]' and we have e^ = -p*(l-p)A^, ^,p = -p»(l-p)4_,. But [1 — p*] = 1 + p + p* = 0. Hence p* = 1, and therefore (1 -p)(l -p'Xl -p*)(l -p')-(l -p)«(l -p')'=9. Hence <'sP = -i^p'(l-p) = 4(l-p')=l^(2+p), whence and the partial fraction is 1 2-hx 1621+a? + a'»' which is ""162 1-a^ ' and gives rise to the prime circulator t^ (2, —1. — l)pcr 3,. The reduction 0ip is somewhat less simple; we have = ^(l-p')(51-10p-14p0 = ^ (61 + 4p - 65p') ; 154] SUPPLEMENTARY BESEABCHES ON THE PARTITION OF NUMBERS. 511 hence finally ^^P = m^-''^ '^ ^^f'^' ^i« = 3^(42 + 23a:); and the partial fraction ia which is J_ . 42 + 23a! S2* '1+x + a/" 1 - 42-19a;-23g' 324 '^^ r^ and gives rise to the prime circulator ssi. ? (42, — 19, — 23) per 3 324 The part depending on the denominator 1 — a; is ^-' +1^0.^^ + A (^e.)* ^ l-x 1 "l-a; ■ 1.2 where l-x 1 ...+ 1.2.3.4.5 (xd,y A. l-x' We have here (1 - O (1 - e-*) (1 - «-") (1 - «-^) (1 - e-«) (1 - e-«) A i . A ^ ,1 1 P = il-. t; + -d_, - ... + j4_, - +&C. logi-3^ = -log« + 2<-24^ + 2880^-'^-' and thence the fraction is , 21, 91„.4S5 720 f which is equal to 720 «• 21 , . 441 ^ . 3087 ^ . 64827 ^ . 1361367 ^ , t + -g-t + ^g- «• + -128" ^ + 1280 "''••• !0 «• /, 21 /, 91^ 8281^ \ 11 7 1 77 1 245 1 . 43981 1 . 199677 1 720 f*^ 480 «• 1080 «« 1152 f 103680 1? 345600 < and consequently the partial fractions are (ae.)» , + ^^^5a (^'Y ^—- + cJ5a («®«)' 1 1 + ooAi (a^*)* 86400 1 - a; ■ 11520 l-x 6480 1-a; ' 2304 103680 from which the non-circulating part is at once obtained. . 43981 ,_^ , 1 ^ + ,,>»^»^ (aW») :; + l-x 199577 1 1 - a;^ 345600 l-a;' 512 SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. [154 The complete expression for the number of partitions is P (1, 2, 3, 4, 5, 6) q = ^QgggQQ (IV + 63(V + 12303» + 1102503» + 4398109 + 698731) + ^ (69' + 126g + 581)(l. 1) per 2, + Jg2? (2, - 1, - 1) per 3, + 3^4 (42, -19. -23) per 3, + 32 (1, 1, - 1, - 1) per 4, + ^ (2, 1, 0, -1.-2) per 5, + ^ ..(2. 1,-1, -2, -1,1) per 6^ 155] 518 155. A FOURTH MEMOIR UPON QUANTICS. [From the Philosophical Transactions of the Royal Society of London, vol. cxlviil fir the year 1858, pp. 415—427. Received February 11,— Read March 18, 1858.] The object of the present memoir is the fiirther development of the theory of binary quantics ; it should therefore have preceded so much of my third memoir, t. 147 (1857), p. 627, [144], as relates to ternary quadrics and cubics. The paragraphs are numbered continuously with those of the former memoira The* first three paragraphs, No& 62 to 64, relate to quantics of the general form (♦$a?, y, ...)^i and they are intended to complete the series of definitions and explanations given in Nos. 54 to 61 of my third memoir; Nos. 68 to 71, although introduced in reference to binary quantics, relate or may be considered as relating to quantics of the like general form. But with these exceptions the memoir relates to binary quantics of any order whatever: viz. Nos. 65 to 80 relate to the covariants and invariants of the degrees 2, 3 and 4; Nos. 81 and 82 (which are introduced somewhat parenthetically) contain the explanation of a process for the calculation of the invariant called the Discriminant; Nos. 83 to 85 contain the definitions of the Catalecticant, the Lambdaic and the Canonisant, which are functions occurring in Professor Sylvester's theory of the reduction of a binary quantic to its canonical form ; and Nos. 86 to 91 contain the definitions of certain covariants or other derivatives connected with Bezout's abbreviated method of elimination, due for the most part to Professor Sylvester, and which are called Bezoutiants, Cobezoutiants, &a I have not in the present memoir in any wise considered the theories to which the catalecticant, &c. and the last-mentioned other covariants and derivatives relate; the design is to point out and precisely define the different covariants or other derivatives which have hitherto presented themselves in theories relating to binary quantics, and so to complete, as far as may be, the explanation of the terminology of this part of the subject. 62. If we consider a quantic (a, 6, ...$a?, y, ...)•" c. II. 65 514 A FOURTH MEMOIR UPON QU ANTICS. [l55 and an adjoint linear form, the operative quantic (a, 6, ...$3|, 9,,...)'*, or more generally the operative quantic obtained by replacing in any covariant of the given quantic the facients (a?, y, ...) by the symbols of differentiation (9^. 9,,...) (which operative quantic is, so to speak, a contravariant operator), may be termed the Pro- vector ; and the Provector operating upon any contravariant gives rise to a contra- variant, which may of course be an invariant. Any such contravariant, or rather such contravariant considered as so generated, may be termed a Provectant; and in like manner the operative quantic obtained by replacing in any contravariant of the given quantic the facients (f , 17, ...) by the sjrmbols of differentiation (9jb, 9y, ...) (which operative quantic is a covariant operator), is termed the Cantraprovector ; and the contraprovector operating upon any covariant gives rise to a covariant, which may of course be an invariant. Any such covariant, or rather such covariant considered as so generated, may be termed a Contraprovectant In the case of a binary quantic, (a, 6, ...][a?, y)*", the two theorems coalesce together, and we may say that the operative quantic (a, 6, ...$9y,-9«)"*, or more generally the operative quantic obtained by replacing in any covariant of the given quantic the facients (a?, y) by the symbols of differentiation (9y, — 9,) (which is in this case a covariant operator), may be termed the Provector. And the Provector operating on any covariant gives a covariant (which as before may be an invariant), and which considered as so generated may be termed the Provectant. 63. But there is another allied theory. If in the quantic itself or in any covariant we replace the facients (x, y, ...) by the first derived functions (9fP, 9^, ...) of any con- travariant P of the quantic, we have a new function which will be a contravariant of the quantic. In particular, if in the quantic itself we replace the &cient8 (a?, y, ...) by the first derived functions (9fP, 9^,...) of the Reciprocant, then the result will contain as a &ctor the Reciprocant, and the other factor will be also a contravariant. And similarly, if in any contravariant we replace the facients (f, 17,...) by the first derived functions (9«TF, dyW,..,) of any covariant W (which may be the quantic itself) of the quantic U, we have a new function which will be a covariant of the quantia And in particular if in the Reciprocant we replace the facients (f, 17, ...) by the first derived functions (dxU, dyU, ...) of the quantic, the result will contain IT as a fisu;tor, and the other factor will be also a covariant. In the case of a binary quantic (a» 6, ...$«, y)* the two theorems coalesce and we have the following theorem, viz. if in the quantic U or in any covariant the facients (x, y) are replaced by the first derived functions (^yWy — 9«TF) of a covariant TT, the result will be a covariant ; and if in the quantic 155] A FOURTH MEMOIR UPON QU ANTICS. 515 U the facients (a?, y) are replaced by the first derived functions (9yJ7, —dxU). of the quantic, the residt will contain ^ as a factor^ and the other fiEU)tor will be also a covariant. Without defining more precisely, we may say that the function obtained by replacing as above the facients of a covariant or contravariant by the first derived functions of a contravariant or covariant is a Transmutant of the first-mentioned covariant or contra- variant. 64. Imagine any two quantics of the same order, for instance, the two quantics F==(a', 6',...$a?, y ...)'", then any quantic such as \U-hfiV may be termed an Intermediate of the two quantics; and a covariant of \U+fiV, if in such covariant we treat \, /i as facients, will be a quantic of the form where the coeflScients (A, B, ... R, A") will be covariants of the quantics U, F, viz. A will be a covariant of the quantic U alone ; J." will be the same covariant of the quantic V alone, and the other coefficients (which in reference to A, A^ may be termed the Connectives) will be covariants of the two quantics; and any coefficient may be obtained from the one which precedes it by operating on such preceding coefficient with the combinantive operator a'3a+6'96 + ..., or fi-om the one which succeeds it by operating on such succeeding coefficient with the combinantive operator ad a' + idb' + • • . > the result being divided by a numerical coefficient which is greater by unity than the index of fi or (as the case may be) \ in the term corresponding to the coefficient operated upon. It may be added, that any invariant in regard to the fiaicients (X, fi) of the quantic (A, B, ... R, A'^\ fiY is not only a covariant, but it is also a combinant of the two quantics 17, F. As an example, suppose the quantics IT, F are the quadrics (a, 6, c\x, yy and (a\ h\ c'\x, yf, then the quadrinvariant of XlT+ZiF is (Xa + fta'XXc + /ic') - (X6 + /iftO*. which is equal to {ac-}?, ac'-2hV + ca\ a'c'-6'«5x, /*)», and oc' — 266' + ca' is the connective of the two discriminants oc — 6* and aV — 6'*. 65—2 516 A FOURTH MEMOIB UPON QUANTICS. [l55 65. The law of reciprocity for the number of the invariants of a binary quantic\ leads at once to the theorems in regard to the number of the quadrinvariantSy cabin- variants and quartinvariants of a binary qmmtic of a given degree, first obtained by the method in the second part of my original memoir*. Thus a quadric has only a single invariant, which is of the degree 2 ; hence, by the law of reciprocity, the number of quadrinvariants of a quantic of the order m is equal to the number of ways in which m can be made up with the part 2, which is of course unity or zero, according as m is even or odd. And we conclude that The quadrinvariant exists only for quantics of an even order, and for each such quantic there is one, and only one, quadrinvariant. 66. Again, a cubic has only one invariant, which is of the degree 4, and the number of cubinvariants of a quantic of the degree m is equal to the number of ways in which m can be made up with the part 4. Hence A cubinvariant only exists for quantics of an evenly even order, and for each such quantic there is one, and only one, cubinvariant. 67. But a quartic has two invariants, which are of the degrees 2 and 3 respectively, and the number of quartinvariants of a quantic of the degree m is equal to the number of ways in which m can be made up with the parts 2 and 3. When m is even, there is of course a quartinvariant which is the square of the quadrinvariant, and which, if we attend only to the irreducible quartinvariants, must be excluded from consideration. The preceding number must therefore, when m is even, be diminished by unity. The result is easily found to be Quartinvariants exist for a quantic of any order, even or odd, whatever, the quadric and the quartic alone excepted; and according as the order of the quantic is 6(7, 6(7 + 1, 6(7 + 2, 6^ + 3, 6(7 + 4, 6(7 + 5, the number of quartinvariants is g, 9 f 9 » 5^ + 1, 9 > 5^+1- In particular, for the orders 2, 3, 4, 5; 6, 7, 8, 9, 10, 11; 12, &a, the numbers are 0, 1, 0, 1; 1, 1, 1, 2, 1, 2; 2, &c Thus the ninthic is the lowest quantic which has more them one quartinvariant. 68. But the whole theory of the invariants or covariants of the degrees 2, 3, 4 is most easily treated by the method above alluded to, contained in the second part of my original memoir; and indeed the method appears to be the appropriate one for the I Introduistoiy Memoir, [189], No. 20. * Ibid. No8. 10-17. 1 i 155] A FOUKTH MEMOIR UPON QUANTIC8. 517 treatment of the theory of the invariants or covariants of any given degree whatever, although the application of it becomes difficult when the degree exceeds 4. I remark, in regard to this method, that it leads naturally, and in the first instance, to a special class. of the covariants of a system of quantics, viz. these covariants are linear functions of the derived functions of any quantic of the system. (It is hardly necessary to remark that the derived functions referred to are the derived functions of any order of the quantic with regard to the facients.) Such covariants may be termed tantipartite covariants; but when there are only two quantics, I use in general the term lineo-linear. The tantipartite covariants, while the system remains general, are a special class of covariants, but by particularizing the system we obtain all the covariants of the par- ticularized system. The ordinary case is when all the quantics of the system reduce themselves to one and the same quantic, and the method then gives all the covariants of such single quantic. And while the order of the quantic remains indefinite, the method gives covariants (not invariants); but by particularizing the order of the quantic in such manner that the derived functions become simply the coefficients of the quantic, the covariants become invariants: the like applies of course to a sjrstem of two or more quantics. 69. To take the simplest example, in seeking for the covariants of a single quantic Uj we in fact have to consider two quantics U, V, An expression such as 12 IT 7" is a lineo-linear covariant of the two quantics; its developed expression is which is the Jacobian. In the particular case of two linear functions (a, 6][a?, y) and (of, V^x, y)y the lineo-linear covariant becomes the lineo-linear invariant ab' — a% which is the Jacobian of the two linear fiinctiona In the example we cannot descend from the two quantics U,VU> the single quantic U (for putting F= 17 the covariant vanishes); but this is merely accidental, as appears by considering a diflferent lineo-linear covariant 12* [/"F", the developed expression of which is In the particular case of two quadrics (a, 6, c$x, yf, (a\ b', (/'^x, y)\ the lineo-linear covariant becomes the lineo-linear invariant ac' - 266' + ca'. If we have V—U, then the lineo-linear covariant gives the quadricovariant d»^U.dy^U^(dJdyUy of the single quantic U (such quadricovariant is in &ct the Hessian) ; and if in the last- mentioned formula we put for U the quadric (a, 6, c][, x, yf, or what is the same thing, if in the expression of the lineo-linear invariant ac' — 266' + ca', we put the two quadrics equal to each other, we have the quadrinvariant ac— 6* of the single quadric. 518 A FOURTH MEMOIR UPON QUANTICS. [155 70. The lineo-lineajr invariant ah* — o!h of two linear functions may be considered as giving the lineo-Iinear covariant d^U .dyV—dyU . dgV of the two quantics U and F, and in like manner the lineo-linear invariant ac' — 2bb' + ca* may be considered as giving the lineo-linear covariant dx^U .dy^V—^dJdyU .dJdyV+dy^U .dx*V o{ the quantics U, V. And generally, any invariant whatever of a quantic or quantics of a given order or orders leads to a covariant of a quantic or quantics of any higher order or orders: viz. the coefficients of the original quantic or quantics are to be replaced by the derived functions of the quantic or quantics of a higher order or ordera 71. The same thing may be seen by means of the theory of Emanants. In tact, consider any emanants whatever of a quantic or quantics; then, attending only to the facients of emanation, the emanants will constitute a system of quantics the coefficients of which are derived functions of the given quantic or quantics; the invariants of the system of emanants will be functions of the derived functions of the given quantic or quantics, and they will be covariants of such quantic or quantics; and we thus pass from the invariants of a quantic or quantics to the covariants of a quantic or quantics of a higher order or orders. 72. It may be observed also, that in the case where a tantipartite invariant, when the several quantics are put equal to each other, does not become equal to zero, we may pass back from the invariant of the single quantic to the tantipartite invariant of the system ; thus the lineo-linear invariant ac* — 266' + ca' of two quadrics leads to the quadiin- variant ac — 6* of a single quantic ; and conversely, from the quadnnvariant oc — 6* of a single quadric, we obtain by an obvious process of derivation the expression cuf — 266' + ca' of the lineo-linear invariant of two quadrics This is in fact included in the more general theory explained. No. 64. 73. Reverting now to binary quantics, two quantics of the same order, even or odd, have a lineo-linear invariant. Thus the two quadrics (a, 6, c^x, yy, (a\ h\ c'^a?, yf have (it has been seen) the lineo-linear invariant acf - 266' + ca* ; and in like manner the two cubics (a, 6, c, djir, y)», (a', 6', c', d'$a:, yf have the Uneo-Unear invariant ad! - 36c' + 3c6' - da\ which examples are sufficient to show the law: 74. The lineo-linear invariant of two quantics of the same odd order is a combinant, but this is not the case with the lineo-linecur invariant of two quantics of the same even order. Thus the last-mentioned invariant is reduced to zero by each of the operations 155] A FOURTH MEMOIR UPON QUANTICS. 519 and but the invariant is by the operations and reduced respectively to and afda + b'db + c'dc + cfda ; ac' -266' 4- ca' a'da + Vdb + c'dc 2(ac - 6^ 2(aV-6'0. 75. For two quantics of the same odd order^ when the quantics are put equal to each other, the lineo-linear invariant vanishes; but for two quantics of the same even order, when these are put equal to each other, we obtain the quadrinvariant of the single quantia Thus the quadrinvariant of the quadric (a, 6, c^x, yY is oc — 6*; and in like manner the quadrinvariant of the quartic (a, 6, c, (2, e\x, yY is oe - 46d + 3c». 76. When the two quantics are the first derived functions of the same quantic of any odd order, the lineo-linear invariant does not vanish, but it is not an invariant of the single quantic. Thus the Uneo-linear invariant of (a, 6, c$a?, yy and (6, c, d^x, yy is (ad — 26c -f c6 = ) od — 6c, which is not an invariant of the cubic (a, 6, c, d$a?, y)". But for two quantics which are the first derived functions of the same quantic of any even order, the lineo-linear invariant is the quadrinvariant of the single quantic. Thus the lineo-linear invariant of (a, 6, c, d^x, yy and (6, c, d, e^x, yy is (a« - 36d + 3c" - d6 =) a« - 4f6(i + 3c*, which is the quadrinvariant of the quartic (a, 6, c, d, e$x, yy. 520 A FOURTH MEMOIR UPON QUANTICS. [155 77. I do not stop to consider the theory of the lineo-linear covariants of two quanties, but I derive the quadricovariants of a single quantic directly fix>m the quadrinvariant. Imagine a quantic of any order even or odd. Its successive even emanants will be in regard to the facients of emanation quantics of an even order, and they will each of them have a quadrinvariant, which will be a quadricovariant of the given quantic. The emanants in question, beginning with the second emanant, are (in regard to the facients of the given quantic assumed to be of the order m) of the orders m — 2, m — 4,... down to 1 or 0, according as m is odd or even, or writing successively 2p+l and 2p in the place of m, and taking the emanants in a reverse order, the emanants for a quantic of any odd order 2p-f 1 are of the orders 1, 3, 5... 2p — 1, and for a quantic of any even order 2p, they are of the orders 0, 2, 4 ... 2p— 2. The quadricovariants of a quantic of an odd order 2p + 1, are consequently of the orders 2, 6, 10...4p — 2, and the quadricovariants of a quantic of an even order 2p, are of the orders 0, 4, 8 ... 4p — 4. We might in each case carry the series one step further, and consider a quadricovariant of the order 4p -f 2, or (as the case may be) 4p, which arises from the 0th emanant of the given quantic; such quadricovariant is, however, only the square of the given quantic. 78. In the case of a quantic of an evenly even order (but in no other case) we have a quadricovariant of the same order with the quantic itsel£ We may in this case form the lineo-linear invariant of the quantic and the quadricovariant of the same order: such lineo-linear invariant is an invariant of the given quantic, and it is of the degree 8 in the coefficients, that is, it is a cubinvariant. This agrees with the before-mentioned theorem for the number of cubinvariants. 79. In the case of the quartic (a, 6, c, d, e$a?, y)*, the cubinvariant is, by the preceding mode of generation, obtained in the form c(ac- 6») -4dJ(ad- 6c)-f 6cJ (oe -46d + 3c«)- 46| (be-cd) + a(ce- (?), which is in ^t equal to 3 (ace -acP- b^e -f 2bcd - c») ; and omitting the numerical factor 3, we have the cubinvariant of the quartic • 80. In the case of a quantic of any order even or odd, the quadrinvariants of the quadricovariants are quartinvariants of the quantic. But these quartinvariants are not all of them independent, and there is no obvious method grounded on the preceding mode of generation for obtaining the number of the independent (asyzygetic) quartin- variants, and thence the number of the irreducible quartinvariants of a quantic of a given order. 81. I take the opportunity of giving some additional developments in relation to the discriminant of a quantic (a, 6, ...6\ a'][a?, y)'». To render the signification perfectly definite, it should be remarked that the discrimiuant contains the term a"*^^a"*^^ and that the coefficient of this term may be taken to be 155] A FOURTH MEMOIB UPON QUANTICS. 521 + 1. It was noticed in the Introductory Memoir, that, by Joachimstharjs theorem, the discriminant, on putting a = 0, becomes divisible by 6', and that throwing out this factor it is to a numerical factor pris the discriminant of the quantic of the order (w— 1) obtained by putting a = and throwing out the factor x] and it was also remarked, that this theorem, combined with the general property of invariants, afforded a convenient method for the calculation of the discriminant of a quantic when that of the order immediately preceding is known. Thus let it be proposed to find the discriminant of the cubic (a, b, c, d^x, yy. Imagine the discriminant expanded in powers of the leading coefficient a in the form then this function. gu^l invariant must be reduced to zero by the operation 369a + 2cdb +dde; or putting for shortness V = 2<^t + dde, the operation is V + Sbda, and we have a^VA+aVB +V(7l ^ = 0, -f a 664 -f SbBj and consequently But C is equal to 6* into the discriminant of (36, 3c, d^x, y)*, that is, its value is 6'(126(i— 9c'), or throwing out the factor 3, we may write C = 46»d-36»c«; this gives £ = - ^^(- 66»cd + 246»cd- 126c*), or reducing ^ = -66cd + 4c»; and thence il = - ^ (- 66d* + Uc'd - 12c»d), or reducing A=d\ which verifies the equation VA = 0, and the discriminant is, as we know, a«d« - 6a6cd -f 4ac» + 46»d - 36V. 82. If we coiisider the quantic (a, 6, ...a$jr, 1)** as expressed in terms of the roots in the form a(x '-ay)(x - /3i/)..., then the discriminant (= a'""^ a *""* + &c. as above) is to a factor prh equal to the product of the squares of the differences of the roots, and the factor may be determined as follows: viz. denoting by f(a, /S, ...) the product of the squares of the differences of the roots, we may write a*"-» ? (a, 13, . . . ) = -AT (d^^ a'"*"* + &c.), c. II. G6 522 A FOUBTH MEMOIB UPON QUANTIC8. [155 where ^ is a number ; and then considering the equation a;*" — 1 = 0, we have to determine N the equation But in general and if then or here and therefore but or and whence or and consequently {:(«. /9, .;.) = (-)•»-■ .V. <f>x = (x — a) (x — 0) ..., (a-/3)(a-7)... = <^'a, &c., ?(a, /3, ...) = (-)*"*<"'-'» f«f/3...; (-ra/87... = -l, a/37...=(-)«-U, J^ = (-)4nr(m-i) ^,n^ a*"-* ?(«, /9, . ; .) = (-)*'"'"»->' m'" (a"*-» a "»-' + &c.), or what is the same thing, the value of the discriminant D (=a'*"*a'*~* + &c.) is (-)Jm(m-i)^-mam-jf(a, /8, ...)• It would have been allowable to define the discriminant so as that the leading term should be m in which case the discriminant would have constantly the same sign as the product of the squared differences; but I have upon the whole thought it better to make the leading term of the discriminant always positive. 83. A quautic of an even order 2p has an invariant of peculiar simplicity, viz. the determinant the terms of which are the coefficients of the pih differential coefficients, or derived functions of the quantic with respect to the facients ; such invariant may also be considered as a tantipartite invariant of the pih emanaubi. Thus the sextic (a, 6, c, (/, e, /, flr$d?, yY 155] A FOUBTH MEMOIR UPON QU ANTICS. 523 has for one of its invariante, the determinant \ a, b, c, d , \ b, Cy d, e \ c , cf , e , f d^ ey f , g The invariant in question is termed by Professor Sylvester the Catalecticaut. 84. Professor Sylvester also remarked, that we may from the eatalecticant form a function containing an indeterminate quantity X, such that the coefficients of the diflFerent powers of X are invariants of the quantic; thus for the sextic, the function in question is a y b ,c ,d— \ b , c , cf + JX, c , c , d - JX, e , / d+\ e , f y g where the la»v of formation is manifest; the terms in the sinister diagonal are modified by annexing to their numerical submultiples of X with the signs + and — alternately, and in which the multipliers are the reciprocals of the binomial coefficients. The function so obtained is termed the Lambdaie, 85. If we consider a quantic of an odd opder, and form the eatalecticant of the ])enultimate emanant, we have the covariant termed the Canonisant. Thus in the case of the quintic (a, 6, c, dy ey f \x, y)\ the canonisant is j ax-{-bg, bx + cy, cx + dy \ \ bx -{- eg, cx-^dyy dx-^-eg .\ ' ex H- dy, dx+ «y, ex -vfg which is equivalent to a , 6 , c , d by C y d y € C , d y e y / , and a like transformation exists with respect to the canonisant of a quantic of any odd order whatever. The canonisant and the lambdaie (which includes of course the • eatalecticant) form the basis of Professor Sylvesters theory of the Canonical Forms of quantics of an odd and an even order respectively. 66—2 524 A FOURTH MEMOIR UPON QUANTIC8. [l55 86. There is another family of covariants which remains to be noticed. Consider any two quantics of the same order, (a, 6,...$a?, y)"», (a, 6', ...$a?, y)~, and join to these a quantic of the next inferior order, where the coefficients {u, v, ...) are considered as indeterminate, and which may be spoken of as the adjoint quantic. Take the odd iineo-iinear covariants (viz. those which arise from the odd emanants) of the two quantics; the term arising from the (2i + l)th emanants is of the form where (-4, jB, ...) are lineo-lineai* functions of the coefficients of the two quantics. Take also the quadricovariants of the adjoint quantic; the term arising from the (2i — m)th emanant is of the form where {U, F, ...) are quadric functions of the indeterminate coefficients (u, v, ...). We may then form the quadrin variant of the two quantics of the order 2(m — 1 — 2i): this will be an invariant of the two quantics and the adjoint quantic, lineo-linear in the coefficients of the two quantics and of the degree 2 in regard to the coefficients (m, v, ...) of the adjoint quantic; or treating the last-mentioned coefficients as iacient«, the result is a lineo-linear m-ary quadric of the form (a, aB,...$u, r,...)«, viz. in this expression the coefficients J3[, J3, ... are lineo linear functions of the co- efficients of the two quantics. And giving to % the different admissible values, viz. from t = to i = ^m — 1 or ^^(m — 1) — 1, according as m is even or odd, the number of the functions obtained by the preceding process is Jm or ^ (m — 1), according as m is even or odd. The functions in question, the theory of which is altogether due to Professor Sylvester, are termed by him Cohezoutiants ; we may therefore say that a cobezoutiant is an invariant of two quantics of the same order m, and of an adjoint quantic of the next preceding order w — 1, viz. treating the coefficients of the adjoint quantic as the facients of the cobezoutiant, the cobezoutiant is an 7H-ary quadric, the coefficients of which are lineo-linear functions of the coefficients of the two quantics, and the number of the cohezoutiants is ^i or ^(m — 1), according as m is even or odd. 87. If the two quantics are the differential coefficients, or first derived functions (with respect to the facients) of a single quantic (a, 6, ...$a?, y)*~. 155] A FOUBTH MEMOIR UPON QUANTICS. 525 then we have what are termed the Cohezoutoids of the single quantic, viz. the cobe- zoutoid is au invariant of the single quantic of the order m, and of an adjoint quantic of the order (m — 2) ; and treating the coefficients of the adjoint quantic as facients, the cobezoutoid is an (wi — l)ary quadric, the coefficients of which are quadric functions of the coefficients of the given quantic. The number of the cobezoutoids is ^(m — 1) or ^ (m — 2), according as m is odd or even. 88. Consider any two quantics of the same order, (a, . . .$a?, y)*~, (a', ...v$a?, y)*", and introducing the new facients (X, F), form the quotient of determinants, (a, ...$0?, y r, {a,.,.^x , y) ^ , y X,Y which is obviously an integral function of the order {m — 1) in each set of facients separately, and lineo-linear in the coefficients of the two quantics; for instance, if the two quantics are (a, 6, c, d^x, yy, {a\ h\ c\ d'^x, yY, the quotient in question may be written ( 3 (at' - a'b\ 3 [ac' - a'c) , ad' - a!d \x, y)« {X, Y)\ 3 {ac' - a'c), ad' - a'd + 9 Q>c' - 6'c), 3 {bd' - b'd) ad'-a'd, Sibd'-b'd) , 3 {cd' ^ c'd) The function so obtained may be termed the Bezoutic Emanant of the two quantics. 89. The notion of such function was in fact suggested to me by Bezout s abbre- viated process of elimination, viz. the two quantics of the order m being put equal to zero, the process leads to (w — 1) equations each of the order (m — 1): these equations are nothing else 'than the equations obtained by equstting to zero the coefficients of the different terras of the series {X, F)*^* in the Bezoutic emanant, and the result of the elimination is consequently obtained by equating to zero the determinant formed with the matrix which enters into the .expression of the Bezoutic emanant. In other words, this determinant is the Resultant of the two quaiitic& Thus lAie resultant of the last-mentioned two cubics is the determinant 3(a6'-ai), 3(ac'-a'c) , ad'-a'd 3(ac'-a'c), ad! - a'd + 9 (be' - b'c), 3(6d'-6'rf) ad'-a'd, 3bd'-b'd , 3(cd'-c'rf) 526 A FOURTH MEMOIR UPON QUANTICS. [155 90. If the two qualities are the dififerential coeflScients or first derived functions (with respect to the facients) of a single qiiantic of the order m, then we hare in like manner the Bezoutoidal Emanant of the single quantic; this is a function of the order (m — 2) in each set of facients, and the coefficients whereof are quadric functions of the coeflBcients of the single quantic. Thus the Bezoutoidal emanant of the quartic (a, 6, c, d, e\x, yY IS ( 3(ac-6»), 3 (ad -be) , ae-bd 5^, y)^(X, Yf 3(ad-6c), ae + Sbd-Qd', S(b€-cd) ae — bd, 3(b€ — cd) , 3 (ce - d* ) , and of course the determinant formed with the matrix which enters into the expression of the Bezoutoidal Emanant, is the discriminant of the single quantic. 91. Professor Sylvester forms with the matrix of the Bezoutic emanant and a set of m facients («i, v, ...) an m-ary quadric function, which he terms the Bezoutiant Thus the Bezoutiant of the before- mentioned two cubics is i 3 {aV - a'6), 3 {ac' - a'c) , ad' - a'd ^w, v, wf ; i 3(ac'-ac), ad' - a'd 4- 9 (6c' - 6'c), 3(6d'-6'd) I ad' - a'd , 36d' - 6'd ,3 (cd' - c'd) , and in like manner with the Bezoutoidal emanant of the single quantic of the order wi and a set of (m — 1) new facients (w, v, ...), an (m— l)ary quadric function, which he terms the Bezoutoid. Thus the Bezoutoid of the before-mentioned quartic is ( 3 (ac — 6'), 3 (ad — be) , ae — bd $w, v, w)\ 3 (ad - 6c), ae 4- 86d - 9c», 3 {be- cd) ae —bd, 3{be — cd), 3 (ce — d*) To him also is due the important theorem, that the Bezoutiant is an invariant of the two quantics of the order m and of the adjoint quantic (u, v, ...$y, — a:)**~S being in fact a linear function with mere numerical coefficients of the invariants called Cobe- zoutiants, and in like manner that the Bezoutoid is an invariant of the single quantic of the order m and of the adjoint quantic {u, v, ...$y, — a?)"*"*, being a linear function with mere numerical coefficients of the invariants called Cobezoutoids. The modes of generation of a covariant are infinite in number, and it is to be anticipated that, as new theories arise, there will be frequent occasion to consider new processes of derivation, and to single out and to define and give names to new co- variants. But I have now, I think, established the greater part by far of the definitions which are for the present necessary. 1561 527 156. A FIFTH MEMOIR UPON QUANTICS. [From the Philosophical Transactions of the Royal Society of London, vol. cxLViii. for the year 1858, pp. 429—460. Received February 11,— Read March 18, 1858.] The present memoir was originally intended to contain a development of the theories of the covariants of certain binary quantics, viz. the quadric, the cubic, and the. quartic; but as regards the theories of the cubic and the quartic, it was found necessary to consider the case of two or more quadrics, and I have therefore com- prised such systems of two or more quadrics, and the resulting theories of the harmonic relation and of involution, in the subject of the memoir; and although the theory of homography or of the anharmonic relation belongs rather to the subject of bipartite binary cjuadrics, yet from its connexion with the theories just referred to, it is also considered in the memoir. The paragraphs are numbered continuously with those of my former memoirs on the subject : Nos. 92 to 95 relate to a single quadric ; Nos. 96 to 114 to two or more quadrics, and the theories above referred to; Nos. 115 to 127 to the cubic, and Nos. 128 to 145 to the quartic. The several quantics are considered as expressed not only in terms of the coefficients, but also in terms of the roots, — and I consider the question of the determination of their linear factors, — a question, in effect, identical with that of the solution of a quadric, cubic, or biquadratic equation. The expression for the linear factor of a quadric is deduced from a well-known formula; those for the linear factors of a cubic and a quartic were *fir8t given in my " Note ' sur les Covariants d'une fonction quadratique, cubique ou biquadratique k deux indc^termin^es," Crelle, vol. L. (.1855), pp. 285 — 287, [135]. It is remarkable that they are in one point of yiew more simple than the expression for the linear factor of a quadric. 92. In the case of a quadric the expressions considered are (a, 6, cjx, y)\ (1) ac-6« , . (2) 528 A FIFTH MEMOIR UPON QU ANTICS. [l56 where (1) is the quadric, and (2) is the discriminant, which is also the quadrin variant, catalecticant, and Hessian. And where it is convenient to do so, I write (2) = D. 93. We have (Be, - 36, da'lx, yy D = tr, which expresses that the evectant of the discriminant is equal to the quadric ; (a, 6, cja^, -a^)« 1/^=40, which expresses that the provectant of the quadric is equal to the discriminant ; (a, 6, c^bx + cy, —ax — hyf = D tT, which expresses that a transmutant of the quadric is equal to the product of the quadric and the discriminant. 94. When the quadric is expressed in terms of the roots, we have a-^ U = (x-ay)(X'-fiy\ and in the case of a pair of equal roots, a''U^(x-OLyy, D =0. 95. The problem of the solution of a quadratic equation is that of finding a linear factor of the quadric. To obtain such linear factor in a symmetrical form, it is necessary to introduce arbitrary quantities which do not really enter into the solution, and the form obtained is thus in some sort more complicated than in the like problem for a cubic or a quartic. The solution depends on the linear transformation of the quadric, viz. if we write (a, 6, c$\x + fiy, vx + pyy = (a', h\ c'Ja?, y)*, 80 that a'— (a, 6, c$X, v)\ V = (a, 6, c$X, i/J/A, p), c' = (a, 6, c\fi, p)», then a'c - 6'« = {ac - ¥) (\p - fipy, an equation which in a different notation is (a, 6, cja:, y)«.(a, 6, c$Z, Yy-{(a, h c^x, yJZ, F))^ = D (F*-Xy)«. 156] A FIFTH MEMOIR UPON QUANTICS. 529 in which form it is a theorem relating to the quadne and its first and second emanant& The equation shows that (a, h, clx, ylX, F) + V rn ( Kr - Zy), where (X, Y) are treated as supernumerary arbitrary constants, is a linear factor of (a, 6, c\xy yy, and this is the required solution. 96. In the case of two quadrics, the expressions considered are (a, 6, c'^x, y)\ (1) (a', h\ c'\x, y)\ (2) ac-h" , (3) ac'-266'4-ca', (4) 6" , (5) ac > » (6) (o6' - a'b, (Xa + fia', ac —ac be' - Vc \x. yf, \c+nc' ^x, yy, {ac -b* , ac'- 266' + ca', a'c' - 6'» \\, ft)', (7) (8) (9) (1) and (2) are the quadrics, (3) and (5) are the discriminants, and (4) is the lineo- linear invariant, or connective of the discriminants ; (6) is the resultant of the two quadrics, (7) is the Jacobian, (8) is an intermediate, and (9) is the discriminant of the intermediate. And where it is convenient to do so, I write (1) = u, (2) = v. (3) = D, (*) = Q. (5) = □'. (6) = R. (7) = H. (8) = W. (9) = e. C. II. 530 A FIFTH MEMOm UPON QUANTIC8. 97. The Jacobian (7) may also be written in the form a, b , c The Resultant (6) may be written in the form [156 a, 26. a, 26, c. a\ 26'. a', 26', c'. and also, taken negatively, in the form 4 {ab' - a'6) (6c' - Vc) - {ac' - a'c)», which is the discriminant of the Jacobian ; and in the form 4 (oc - ¥) (a V - 6'») - {ac' - 2hV + caj, which is the discriminant of the Intermediate. 98. We have the following relations: (a, 6, c\Vx + c'y, - a'x - 6'y)» = - (aV - 6'*) (a , 6 , c\x, yf + (ac'-266'4-ca') (a', 5', c'$a?, y)«, (a', 6', c' \hx -\-cy,-ax- byY = + (oc' - 266' + ca') (a , 6 , c $a?, y)» -(ac-6«) (a', 6', c'l^x, y)\ and moreover (ac-6«, ac' -266' 4- ca', a'c'-6'«$£r, - tO* = - {{ab' - a'6, oc' - a'c, 6c' - 6'c][a', y)*}*, an equation, the interpretation of which will be considered in the sequel. "99. The most important relations which may exist between the two quadrics are First, when the connective vanishes, or ac' - 266' + ca' = 0, in which case the two quadrics are said to be karmonically related: the nature of this relation will be further considered. 156] A FIFTH MEMOIR UPON QU ANTICS. 531 Secondly, when i2 = 0, the two quadrics have in this case a common root, which is given by any of the equations, = 3ii'B : dff'R : d^R = 6c' — Vc : ca' — c'a : aV — a'6. The last set of values express that the Jacobian is a perfect square, and that the two roots are each equal to the common root of the two quadrics. The preceding values of the ratios a^ : 2xy : y' are consistent with each other in virtue of the assumed relation i2 = 0, hence in general the functions 4SaR . dcR - (djty, daR . db'R - dbR . daR, &c. all of them contain the Resultant i2 as a factor. It is easy to see that the Jacobian is harmonically related to each of the quadrics; in fact we have identically a(6c'-6'c)4-6 (ca'-c'a)4-c (a6'-a'6) = 0, a' (6c' - 6'c) + V {ca' - c'a) 4- c' (a6' - a'6) = 0, which contain the theorem in question. 100. When the quadrics are expressed in terms of the roots, we have a-^U =(a?-ay)(a?-^y), a'-'W =(^-a'y)(j:-^'y), 4a-^n =-(a-)8)«, 2 (aa')-' = 2a^ + 20^)8' -{a -^13) (a' + /S'), 4a'-^n' =-(a'-^')», (aa^-^R =(a-a')(a-^')()8-a')(/S-/8'), (oaT'H = y«, 2yx , x^ 1, a+^, afi 1, a' + iS', o^ff 101. The comparison of the last-mentioned value of -K with the expression in terms of the roots obtained from the equation -ij=4nn'-Q», gives the identical equation which may be easily verified. 67—2 532 A FIFTH MEMOIR UPON QUANTIC8. [156 102. We have identically 2a/8 + 2a'/8' - (a + yS) (o' + /S') = 2 (a -«')(« -/SO -(a -/9)(2a = 2(/3-a')(y8 -^)-(fi -a)(2ff = 2(0' -a)(a' - /3 ) - (a' - /T) (2a' = 2 09' - a ) (/y - yS ) - (/3' - 0') (2/3' a: a' a a /9); and the equation Q^^ac' — 266' + ca' = . njay consequently be written in the several forms a-/S a-a' a-fi 2 1 1 / > /3-a a-a' ' fi'-fi' 1 1 + -r a'-/3' a'-a ' o'-/8' 1 1 + so that the roots (a, yS), (a', /3') are harmonically related to each other, and hence the notion of the harmonic relation of the two quadrics. 103. In the case where the two quadrics have a common root a = a', a-' U = (^- ay) {x - -/8y), a'-' U' = (x- ay) {x - -/9'y). 4a-*n = -(a -/8)'. 2 (aa')-'Q = («- |8)(a- ■n 4a'- -,□' = -(a -yS')'. R = 0, (m')-' H = (^ - 0) (X - a^y. 104. In the case of three quadrics, of the expressions which are or might be considered, it will be sufficient to mention (a , b , c Ijix, yY, (a ,h',c' $ar, yf. (a", b", c"\x, yy. a , b , c ' > a , , c a", b", c" (1) (2) (3) (4) 156] A FIFTH MEMOIR UPON QU ANTICS. 533 where (1), (2), (3) are the quadrics themselves, and (4) is an invariant, linear in the coefficients of each quadric. And where it is convenient to do so, I write (1) = u, (2) = W, (3) = U". (4) = n. 105. The equation H = is, it is clear, the condition to be satisfied by the coefficients of the three quadrics, in order that there may be a syzygetic relation \U + fjkU' + vU" = 0, or what is the same thing, in order that each quadric may be an intermediate of the other two quadrics; or again, in order that the three quadrics may be tn Involution. Expressed in terms of the roots, the relation is 1, a +13, afi 1, af+^, a[ff 1, a" + ^", a";8 n Qii = 0; and when this equation is satisfied, the three pairs, or as it is usually expressed, the six quantities % fi\ a!, ff \ ol\ ff\ are said to be in involution, or to form an involution. And the two perfectly arbitrary pairs a, )8; oi^ ff considered as belonging to such a system, may be .spoken of as an involution. If the two terms of a pair are equal, e.g. if a" = ff' = ^, then the relation is 1, 2^ , ^ 1, a+)8, dfi 1, o' + ^S', alff = 0; and such a system is sometimes spoken of as an involution of five terms. Con- sidering the pairs (a, fi\ (a', )8') as given, there are of course two values of 6 which satisfy the preceding equation; and calling these 6^ and 6^^, then 6, and 0,^ are said to be the sibiconjugates of the involution a, ^8; a', ff. It is easy to see that ^^, 6^^ are the roots of the equation H=0, where H is the Jacobian of the two quadrics U and IT whose roots are (ot, /8), («', ff). In fact, the quadric whose roots are 0,y 0„ is y\ 2yx , od" 1, a+)8, ap 1, a' + ^, a'^ which has been shown to be the Jacobian in question. But this may be made clearer as follows: — If we imagine that X, fi are determined in such manner that the inter- mediate \V -¥ fiir may be a perfect square, then we shall have \U'{-fiir — a''{x'-0yY, where denotes one or other of the sibiconjugates 0^, 0^^ of the involution. But the condition in order that \U •{■ filT may be a square is {ac - h\ ad - 266' + ca, dd - y'\\ fiy ; 534 A FIFTH MEMOIR UPON QUANTICS. [156 and observing that the equation \ : fi= W : -^-U implies \U+ filT ^0=^a'' (x— OyY, it is obvious that the function must be to a fetctor prh equal to (x — O^yf (x — 0„yy. But we have identically (ac - 6«, ac' - 266' + ca\ ale' - V'^W, ^U)'^^ {{ab' - a% ac' - a'c, be' - Vcl^x, y)»}«, and we thus see that (x — 0^y), (x — 0^^y) are the £Eu;tors of the Jacobian. 106. It has been already remarked that the Jacobian is harmonically related to each of the quadrics U, U'; hence we see that the sibiconjugates 5,, 0„ of the involution a, y3, a^, ff are a pair harmonically related to the pair a, /3, and also harmonically related to the pair a\ ^, and this properly might be taken as the definition for the sibiconjugates 5,, 0,^ of an involution of four terms. And moreover, a, P\ a\ fi' being given, and 0^, 0^^ being determined as the sibiconjugates of the involution, if a", /S^' be a pair harmonically related to 0,, 5,,, then the three pairs a, /9; a', )8'; a", /8" will form an involution; or what is the same thing, any three pairs a, /8; oi, ^\ a", ^", each of them harmonically related to a pair 0,, 6^^, will be an involution, and 0^ , 0^^ will be the sibiconjugates of the involution. 107, In particular, if a, yS be harmonically related to 0^, 5,^, then it is easy to see that ^^, 0^ may be considered as harmonically related to d^, d^^, and in like manner ^//» ^// wi^' ^^ harmonically related to 5,, 0^/, that is, the pairs 5,, 0/^ 0^^, 0^^ and a, fi will form an involution. This comes to saying that the equation 1, ^e, . e; =0 1, a + /3, «^ is equivalent to the harmonic relation of the pairs a, /S; B^, 0^/, and in &ct the deter- minant is (d, - 0„) {2afi + 20,0,, -(a + 13) (0, + 0,,)), which proves the theorem in question. 108. Before proceeding further, it is proper to consider the equation = 0, 1, a, a', aal 1. fi. ^, ^ff 1. 7. i. r/ 1, 8. 8'. ^ which expresses that the sets (a, yS, 7, S) and (a', ff, y\ S^ are homographic; for although the homographic equation may be considered as belonging to the theory of 156] A FIFTH MEMOIR UPON QU ANTICS. 535 the bipartite quadrie {x — ay) (x — ay), yet the theory of involution cannot be completely discussed except in connexion with that of homography. If we write ^ =(/S-7)(«-8), 5=(7-a)(^-8), C ={a - p)iy -i), A' = (^-Y)(c^-n B' = W-c^)(B'-n (7 = (a' - jSO (7' - 8'), then we have and thence BC-FC^CA''-C'A^AR^A'B; and either of these expressions is in fact equal to the last-mentioned determinant, as may be easily verified. Hence, when the determinant vanishes, we have A : B : C^A' : R : C, Any one of the three ratios A : B : C, for instance the ratio B : C,= (7-«)(/8"g) (a-)8)(7-8)' . is said to be the anharmonic ratio of the set (a, ^, 7, S), and consequently the two sets (a, ^, 7, S) and (a\ ff, 7', S') will be homographically related when the anharmonic ratios (that is, the corresponding anharmonic ratios) of the two sets are equal. If any one of the anharmonic ratios be equal to unity, then the four terms of the set taken in a proper manner in pairs, will be harmonics; thus the etiuation ^ = 1 gives which is reducible to 2aS 4. 2^7 - (a + S) 08 + 7) = 0, which expresses that the pairs a, B and 13, 7 are harmonics. 109. Now returning to the theory of involution (and for greater convenience taking a, 0^ &c. instead of a, )8 &c. to represent the terms of the same pair), the pairs a, a'; )8, iS'; 7, 7'; S, 8'; &c. will be in involution if each of the determinants formed Mrith any three lines of the matrix 1, a +a! , aa' , 1, fi + fi', pff, ^ 7+7» 77'» 1, S+S', SS', &c. 536 A FIFTH MEMOIR UPON QUANTICS. [156 vanishes: but this being so, the determinant which is equal to 1, a, a', aa' 1. A ^. 18/8' 1, 7. 7'. 77' 1, 8, 8', SS' «■ 1, « + «' , aa' /9. 1, 13 + ^ > fi^ 7. 1. 7+7' . 77' B, 1, B +B' , SS' will vanish, or the two sets (a, )8, 7, S) and (a', /S', 7', SO will be homographic ; that is, if any number of pairs are in involution, then, considering four pairs and selecting in any manner a term out of each pair, these four terms and the other terms of the same four pairs form respectively two sets, and the two sets so obtained will be homographic. 110. In particular, if we have only three pairs a, a'; fi, ff\ 7, 7', then the sets a, )8, 7, a' and a', ff, 7', a will be homographic; in fact, the condition of homography is which may be written or what is the same thing, 1, a, a', aa! = } 1, A ^. P^ 1. 7. 7» Ti 1, a'. a, ojoi a. 1, a 4- a', aa' J= /9. 1, /S + i8', P^ 7. 1. 7 + 7'. 77' a'. 1. a +0', ew' a > 1, a + a* , oa' /3 9 1. iS + yS', y9/8' 7 1 1. 7+7'. 77' 0'- -a, 0, . c ) = 0, so that the first-mentioned relation is equivalent to («' - «) 1, a + a' , ao* | = 0, 1, y8 + ^, /SyS- 1. 7+7'. 77' 156] A FIFTH MEMOIR UPON QUANTICS. 537 and the two sets give rise to an involution. The condition of homography as expressed by the equality of the anharmonic ratios may be written a-7.a'-^'"a'-7'.a-/8'' or multiplying out, (a - ;8) (a - ;80 («' - 7) (0' - yO - («' - ;8) (a' - )8') (a - 7) (a' - 7') = 0. which is a form for the equation of involution of the three pairs. But this and the other transformations of the equation of involution is best obtained by a different method, as will be presently seen. Ill, Imagine now any number of pairs a, a'; fi, /3^; 7, 7'; S, S'; &c. in involution, and let x, y, z, w he the fourth harmonics of the same quantity X with respect to the pairs a, a' ; fi, 13" ; 7, 7' and B, 8' respectively ; then the anharmonic ratios of the set (Xy y, z, w) will be independent of X, or what is the same thing, if x\ y\ /, w are the fourth harmonics of any other quantity X' with respect to the same four pairs, the sets (a?, y, ^, w) and (x\ y', /, w') will be homographic, or we shall have = 0. 1, X, x\ OCX 1, y> y'. yy 1, z. «', Z!f 1, w, w', vm It will be sufficient to show this in the ceese where X is anything whatever, but X' has a determinate value, say X' = 00 ; and since if all the terms a, a', &c. are diminished by the same quantity X the relations of involution and homography will not be affected, we may without loss of generality assume X = 0, but in this case X = — — , , a;' = i (a -h a'), and the equation to be proved is 1. aa 1, 1, 1, a + a" 77' / f 7+7 SB' B + B^' a -f flf, aa* ; = 0, /3 + /8', 13^ 7+7'» . 77' B + B\ BB' which is obviously a consequence of the equations which express the involution of the four pairs. C. II. 68 538 A FIFTH MEMOIR UPON QUANTICS. [156 A set homographic with x^ y, z w, which are the fourth harmonics of any quantity whatever X with respect to the pairs in involution, a, a'; fi, 13'; 7, 7'; S, B\ is said to be homographic with the four pairs, and we have thus the notion of a set of single (quantities homographic with a set of pairs in involution. This very important theon- is due to M. Chasles. 112. Let r; 8] t he the anharmonic ratios of a set a, fi, 7, B, and let r/, s,; t, be the anharmonic ratios (corresponding or not corresponding) of a set a^, ^^, 7^, h^ And suppose that /; ^; t'; r/; <; e/; r"; *"; T; <; <'; C; ^'"; «'"; ^"; r/"; <"; C are the analogous quantities for three other pairs of sets ; then an equation such as TV, TV, = 0. or as it is more conveniently written, »«, > »•». . »•/« . »^, «v , rv; A" . »^V' . r/V . «"V", r'V. r/'V", r r, = is a relation independent of the particular ratios r : 8 which have been chosen for the anharmonic ratios of the sets; this is easily shown by means of the equations r + « + « = 0, r, + «, + «, = 0, which connect the anharmonic ratios. The equation in fact expresses a certain relatiou between four sets (a, ^, 7, S) and four other sets (a^, /8^, 7^, S^); a relation which may be termed the relation of the homography of the anharmonic ratios of four and four sets : the notion of this relation is also due to M. Chasles. 113. The general relation 1, a + /8 , a/3 =0 1, ot +^', f£ff 1, a" + i8", ot'P' may be exhibited in a great variety of forms. In fact, if the determinant is denoted by T, then multiplying by this determinant the two sides of the identical equation w', — M, 1 v", -V, 1 V^^ "W, 1 we obtain T (u — v) (v — w) (w — w) = = (w — v) (v — w) (w — u\ (M-a)(t/-/3), (t;-a)(t;-)8), (w-a )(«;- )8 ) !. (^-a')(u-/9^), (t;-a')(t;-/}'), (w^a')(w^^) (^-0(^-/9^), (t^-a'Xt'-rX {wa")(w^l3'') 156] A FIFTH MEMOIR UPON QUANTICS. 539 If, for example, u = a, v = yS, then we have T (a - ;8) = - (a - a') (a - /90 C9 - O (/9 - /8") + (/3 - «') (/3 - ^) (« - «") (« - /3") ; and again, if « = a, v = a', v> = a", then we have T = - (a - /3") (a' - ;8) (a" - i80 + (a - /S-) («' - /9") («" - yS). Putting T = Q, the two equations give respectively (g - a') (ff - c^') _ (« - yyp (/3 - /y) . (a-«")(«'-/8)~(«-i8')(i9"-/9)' and (« - /8") («' - /3) («" - /8') = (a - yS-) («' - r) («" - /8), which are both of them well-known forma 114. A corresponding transformation applies to the equation which expresses the homography o representing by V the similar determinant , a, of, aal =0, , A ff. (iff > 7» i> Tl , &, O y CO two pairs. In fact, calling the determinant 'V and V9 = ss' , — »' , -s, 1 > ttf, -if. -t, 1 uu', — «', -tt. 1 m/ , —v'. -V. 1 ted to zero, would express the we have homography of the sets («, t, II, v) and («-«)(«'-«'). {8-fi)(8'-n («-7)(«'-7). («-«)(*' -80 9 (t - a) (f - a'), {t-fi)(1f -n (t - 7) «' - 7 ). (<-«)(«'-«') (« - a) («' - a!), (u - /S) (u' - ^, (« - 7) («' - y). (m-8)(m'-S0 (v-a){t/-af), (v- dyw-n (» - 7) (» -7). (» - S) (t/ - 8') which gives various forms of the equation of homography. In particular, if « = o^ s' = ff', t = ff, (f = a', u = y, «' = S', » = S, v'=y, then 7^ = (« (/3 7)03' 7) (a' 70, (« 70. (-8 S)08'-80 (7 - «) (S' (8 - a) (y «0. (7 /8)(8' y8)(7' /SO 68—2 540 A FIFTH MEMOIR UPON QUANTICS. [156 and the right-hand side breaks up into factors, which are equal to each other (whence also V = '^), and the equation S?^ = takes the form (« - 7) (/8 - S) («' - S') (/S- - 7') - (« - 8) (-8 - 7) («' - 7') (/S' - «') = 0. which is, in fact, one of the equations which express the equality of the anharmonic ratios of (a, )8, 7, S) and (a', ff, y\ S^. 115. In the case of a cul:)ic, tlie expressions considered are (a, 6, c, d$a?, y)», (ac — b^ ad — be, bd — t^^a;, yf, - a^d + %abc - 26» '\ - abd + 2a(^ - 6«c ■\-acd- 2b^d 4- 6c* [ -\-ad' - Sbcd + 2c* ; a«(? - 6abcd + 4ac» 4- 46»df - 36V, (1) (2) [^> y)** (3) (4) where (1) is tlie cubic, (2) is the quadrico variant or Hessian, (3) is the cubico variant, and (4) is the quartinvariant or discriminant. And where it is convenient to do so, I write (1) (2) (3) (4) U, H. so that we have *»-ni7»4-4JJ» = 0. 116. The Hessian may be written under the form (cur + by)(cx-\-dy) — (6a? 4- cy)*, (which, indeed, is the form imder which qua Hessian it is originally given), and under the form a, b ., c 6 , c , (Z The cubicovariant may be written under the form {2 (ac - 6") a? + (ad - be) y] (6a^ + 2cxy 4 dy») - { {ad-bc)x + 2{bd - c")y} (cue* + 2&py 4- cy*), 156] A FIFTH MEMOIR UPON QU ANTICS. 541 that is, as the Jacobian of the cubic and Hessian ; and under the form that is, as the evectant of the discriminant. The discriminant, taken negatively, may be written under the form + 4 (oc - b^)(bd -'(^)-{ad- hcf, that is, as the discriminant of the Hessian. 117. We have (a, 6, c, d\ha^ + 2ca>y + df, -aaf- ihayy - c}/J = ir<^, which expresses that a transmutant of the cubic is the product of the cubic and the cubicovariant. The equation {(3a, 9^, 3c, 9dl[», -^)'}«n=2Z7' expresses that the second evectant of the discriminant is the square of the cubic. The equation # . -3cd , -36d + 6c« , -36c-h2ad = 27 D' -3cd , -3c« +126d, -3ad-66c , -3ac + 66« -36d-f6c» , -3arf-66c , -36« + 12ac, -3aft I - 36c - 12ad, - 3ac + 66' , 3a6 , a« ' expresses that the determinant formed with the second differential coefficients of the discriminant gives the square of the discriminant. The covariants of the intermediate aU-h /3^ are as follows, viz. 118. For the Hessian, we have H(aU-^l3<P)= (1, 0, -n3[a, /3yH = (a>-/8»n)ir; for the cubicovariant, ^(a[7+^4>)= (0, D, 0, -n» 5a, fifU + (1, 0, -D, 05a,^)»c|> and for the discriminant, Q(a£r + ^4>)= (1, 0, -2D, 0, D^^a, fi)*^ = (a«-/S«n)«D, where on the left-hand sides I have, for greater distinctness, written JV, &c. to denote the functional operation of taking the Hessian, &c. of the operand aU + fi^, 542 A FIFTH MEMOIR UPON QUANTICa [156 In particular, if a = 0, ^ = 1, 119. Solution of a cubic equation. The question is to find a linear factor of the cubic (a, 6, c, d$a?, yY, and this can be at once effected by means of the relation between the covariants. The equation in fact shows that each of the expressions is a perfect cube, and consequently that the cube root of each of these expressions is a linear function of (x, y). The expression is consequently a linear function of x, y, and it vanishes when [7 = 0, that is, the expression is a linear factor of the cubic. It may be noticed here that the cubic being a(a? — ay)(a? — ^y)(a? — 7y), then we may write ^Ji^Tu^) - ^^(^-U^U) = J a(a) -tti>)(^ -7)(a?-ay), where 6> is an imaginary cube root of unity: this will appear firom the expressions which will be presently given for the covariants in terms of the roots. 120. Canonical form of the cubic. The expressions i(4>+ [/VD), i(4>— tTVO) are perfect cubes; and if we write then we have U^ x» + y», 4) = Vn (x» - y»), and thence also ir=-^nxy. 156] A FIFTH MEMOIR UPON QUANTICS. 543 121. When the cubic is expressed in terms of the roots, we have a-^fT^ (x - ay){x - fiy)(x - 7^) ; and then putting for shortness ^=0-ry)(a:-ay), B ^ {r^ -■ a) {x -- fiy\ C = (a - ^){x -yy\ so that Jl+£ + C = 0, we have a-*^ =^^{B''C){C^A){A^B\ 122. The covariants -ff, 4> are most simply expressed as above, but it may be proper to add the equations a" 'a» + /8* + 7* - ^7 - 7a - aA = -ij 6a^7-i87«-7a»-a)3»-)8V-7«a-a«/3, jTo?, y^ = - 4 {(a + ©i8 + ai«7) a? + (^7 + tt)7a + w'^aiS) y 1 {(« + ©'^/S -h 0)7) a- 4- (^7 4- ©V* + ®«i8) 3^] (where © is an imaginary cube root of unity), a-^4>=^2(a-^)(a-7)«(a:-^y)«(ar-7y) ' 2(a« + ^' + 7')-3(^ + 7a' + aiS" + )8»7 + 7*a + o?/3) + 12a^7, -2(a«/87 + /8Va + 7'a^) + 4(^' + 7»a« + a»^)-(i87» + 7a« + a^ + ^7 + y^ - 2 (a^V+)87'a*+7aW+4(a»^7+i8V+7*«i8)-(^+7»aHa«/9'+^^ ' "^ ^+2(^V + 7V + a»/3»)-3(a)8V + )87V + 7a«/3» + a^V + ^7'«' + 7<3^i8") + 12^^ ; = {(2a-^~7)a;+(2^7-7a~a^)^} K2^-7-a)^+(27a -,7^-^7)3^} {(27-a-^);r+(2a^-/87-7a)yj 123. It may be observed that we have a-^UU^ = - ^ A^B^O, which, with the above values of H, 4> in terms of A, B, C and the equation A + B-hG = 0, verifies the equation 4>' — OU* + 4iH^ = 0, which connects the covariants. In fact, we have identically, {B^cyiC-AyiA-By^^ -'4(A-^B + CyABC-^{A+B-hC)^{BC-^CA + ABy-^lS{A+B-^C)(BC-{-CA-{-AB)ABa -^(BC+CA+ABy- 27 A'B'O, by means of which the verification can be at once eflfected. 544 A FIFTH MEMOm UPON QUANTICS. [156 124. If, as before, cu is au imaginary cube root of unity, then we may write 27a-»4> =-(£-C)(C-A)(il~5), 27a-» fT Vn = 3 (ft) - ft)«) ABC, and these values give 27a-»i(4)+ U'JD = [{a + 0)^ + arf) X -^ (By + a)^a + <oa^ ) y}\ 27a-» J (^ - ^ ^3 = ((« + <»^ + ^'t) a? + (^87 + «7a + «'a/8) y}'» and we thence obtain ^J(4>+£rVn) - ^i(4>- [TVd) = _ Ja (ft> - fti')(i8 - 7)(ar - ay)